0321_1032_2103
Counting sequence:
1, 1, 2, 6, 21, 76, 273, 964, 3356, 11587, 39866, 137055, 471326, 1621698, 5581897, 19216642, 66160957, 227786244, 784234694, 2699961637, 9295329610, 32001480693, 110172921570, 379297324332, 1305824900183, 4495627408144, 15477320716567, 53284548958172, 183445393734274, 631556663364173, 2174291780607416, 7485543205564869, 25770854336879534, 88722610223504514, 305449771397643829, 1051587218515638352, 3620351960546055581, 12463966936274142720, 42910322946633613208, 147729517002195601483, 508595803870402733366, 1750968235450752164067, 6028146001563955078090, 20753400021877575118846, 71448769216157646327681, 245980254662945886379078, 846848537041046430479289, 2915487853574559766857112, 10037295989249528856814026, 34555901391356728968841405, 118967331664640591381451342, 409574788488790278257361993, 1410063628547302912363868838, 4854496644893879218309619472, 16712818626180991331533089447, 57538057364900568959511131804, 198089150572149279390306545667, 681971435454344408024643652912, 2347857201832278827642742366614, 8083085527655855295563951946717, 27828043203143196132288388735148, 95804997468656632861964440204633, 329832660994739831956736985415194, 1135531414156785985624033702339838, 3909350846723678577296981100919285, 13458917870737984497639743816415172, 46335690336691294024527854569053625, 159522200788942183491970641119515836, 549195066689163116578735272346481044, 1890741350006636782866238004449248787, 6509349900348368237649649112987650338, 22410064774336574795416047513499914999, 77152251895858848059032154343453300470, 265615920013701589609840936040693802138, 914449225149732717541375491100590961497, 3148220126767292185839786156755045107002, 10838535037262218659679384569944796842597, 37314367173742444106656632860317974831052, 128464039908531229631379500233486873733326, 442269581386164380962016480052306209505717, 1522623628828468964099428229269294881735874, 5242012592863574482131318481720532110271309, 18046939180158950006736385547732401087851338, 62131100985096863203237661422662565738664884, 213901851781289413331289499128761014565587319, 736410613525737023055611464568956013253665560, 2535277685523940606628187641479626226012028255, 8728327409543762527531346198874808253379133220, 30049449732149873399712903575206518238673711498, 103452744934576316159820795293428440369053819405, 356161944058760724341588050991133153876193664656, 1226176555063249321212765228917527773491301799293, 4221419411217959974501677497261364696834394794310, 14533291940562806638547136138673366602737130307258, 50034491732411829151383764782994044726513241548501, 172256249524141928411613930170241198974754189164456, 593035213764389862065118048880231629568116435056037, 2041672019076937346121639337768790079967879527469176, 7028966470678744170304087492402981816937778981458128, 24198974754164075181735426869682153210979115211849371, 83311021840188603500490095009265745601758740162744574
Generating function in Maple syntax:
(x^4-9*x^3+12*x^2-6*x+1)*(-1+x)^4/(5*x^8-44*x^7+128*x^6-208*x^5+209*x^4-132*x^3+51*x^2-11*x+1)
Generating function in latex syntax:
\frac{\left(x^{4}-9 x^{3}+12 x^{2}-6 x +1\right) \left(-1+x \right)^{4}}{5 x^{8}-44 x^{7}+128 x^{6}-208 x^{5}+209 x^{4}-132 x^{3}+51 x^{2}-11 x +1}
Generating function in sympy syntax:
(x - 1)**4*(x**4 - 9*x**3 + 12*x**2 - 6*x + 1)/(5*x**8 - 44*x**7 + 128*x**6 - 208*x**5 + 209*x**4 - 132*x**3 + 51*x**2 - 11*x + 1)
Implicit equation for the generating function in Maple syntax:
(5*x^8-44*x^7+128*x^6-208*x^5+209*x^4-132*x^3+51*x^2-11*x+1)*F(x)-(x^4-9*x^3+12*x^2-6*x+1)*(-1+x)^4 = 0
Implicit equation for the generating function in latex syntax:
\left(5 x^{8}-44 x^{7}+128 x^{6}-208 x^{5}+209 x^{4}-132 x^{3}+51 x^{2}-11 x +1\right) F \! \left(x \right)-\left(x^{4}-9 x^{3}+12 x^{2}-6 x +1\right) \left(-1+x \right)^{4} = 0
Explicit closed form in Maple syntax:
-482750725/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 1)^(-n+6)-482750725/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 2)^(-n+6)-482750725/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 3)^(-n+6)-482750725/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 4)^(-n+6)-482750725/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 5)^(-n+6)-482750725/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 6)^(-n+6)-482750725/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 7)^(-n+6)-482750725/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 8)^(-n+6)+4026054590/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 1)^(-n+5)+4026054590/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 2)^(-n+5)+4026054590/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 3)^(-n+5)+4026054590/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 4)^(-n+5)+4026054590/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 5)^(-n+5)+4026054590/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 6)^(-n+5)+4026054590/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 7)^(-n+5)+4026054590/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 8)^(-n+5)-10481128598/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 1)^(-n+4)-10481128598/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 2)^(-n+4)-10481128598/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 3)^(-n+4)-10481128598/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 4)^(-n+4)-10481128598/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 5)^(-n+4)-10481128598/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 6)^(-n+4)-10481128598/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 7)^(-n+4)-10481128598/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 8)^(-n+4)+15075243493/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 1)^(-n+3)+15075243493/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 2)^(-n+3)+15075243493/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 3)^(-n+3)+15075243493/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 4)^(-n+3)+15075243493/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 5)^(-n+3)+15075243493/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 6)^(-n+3)+15075243493/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 7)^(-n+3)+15075243493/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 8)^(-n+3)-12863997899/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 1)^(-n+2)-12863997899/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 2)^(-n+2)-12863997899/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 3)^(-n+2)-12863997899/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 4)^(-n+2)-12863997899/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 5)^(-n+2)-12863997899/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 6)^(-n+2)-12863997899/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 7)^(-n+2)-12863997899/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 8)^(-n+2)+2122346689/61845461*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 7)^(-n+1)+2122346689/61845461*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 8)^(-n+1)+2122346689/61845461*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 1)^(-n+1)+2122346689/61845461*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 2)^(-n+1)+2122346689/61845461*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 3)^(-n+1)+2122346689/61845461*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 4)^(-n+1)+2122346689/61845461*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 5)^(-n+1)+2122346689/61845461*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 6)^(-n+1)+62557958/61845461*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 1)^(-n-1)+62557958/61845461*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 2)^(-n-1)+62557958/61845461*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 3)^(-n-1)+62557958/61845461*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 4)^(-n-1)+62557958/61845461*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 5)^(-n-1)+62557958/61845461*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 6)^(-n-1)+62557958/61845461*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 7)^(-n-1)+62557958/61845461*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 8)^(-n-1)-1681531585/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 1)^(-n)-1681531585/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 2)^(-n)-1681531585/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 3)^(-n)-1681531585/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 4)^(-n)-1681531585/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 5)^(-n)-1681531585/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 6)^(-n)-1681531585/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 7)^(-n)-1681531585/185536383*RootOf(5*_Z^8-44*_Z^7+128*_Z^6-208*_Z^5+209*_Z^4-132*_Z^3+51*_Z^2-11*_Z+1,index = 8)^(-n)+piecewise(n = 0,1/5,0)
Explicit closed form in latex syntax:
-\frac{482750725 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +6}}{185536383}-\frac{482750725 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +6}}{185536383}-\frac{482750725 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +6}}{185536383}-\frac{482750725 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +6}}{185536383}-\frac{482750725 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +6}}{185536383}-\frac{482750725 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +6}}{185536383}-\frac{482750725 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +6}}{185536383}-\frac{482750725 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +6}}{185536383}+\frac{4026054590 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +5}}{185536383}+\frac{4026054590 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +5}}{185536383}+\frac{4026054590 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +5}}{185536383}+\frac{4026054590 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +5}}{185536383}+\frac{4026054590 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +5}}{185536383}+\frac{4026054590 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +5}}{185536383}+\frac{4026054590 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +5}}{185536383}+\frac{4026054590 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +5}}{185536383}-\frac{10481128598 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +4}}{185536383}-\frac{10481128598 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +4}}{185536383}-\frac{10481128598 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +4}}{185536383}-\frac{10481128598 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +4}}{185536383}-\frac{10481128598 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +4}}{185536383}-\frac{10481128598 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +4}}{185536383}-\frac{10481128598 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +4}}{185536383}-\frac{10481128598 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +4}}{185536383}+\frac{15075243493 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{185536383}+\frac{15075243493 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{185536383}+\frac{15075243493 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{185536383}+\frac{15075243493 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{185536383}+\frac{15075243493 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{185536383}+\frac{15075243493 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +3}}{185536383}+\frac{15075243493 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +3}}{185536383}+\frac{15075243493 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +3}}{185536383}-\frac{12863997899 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{185536383}-\frac{12863997899 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{185536383}-\frac{12863997899 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{185536383}-\frac{12863997899 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{185536383}-\frac{12863997899 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{185536383}-\frac{12863997899 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +2}}{185536383}-\frac{12863997899 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +2}}{185536383}-\frac{12863997899 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +2}}{185536383}+\frac{2122346689 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +1}}{61845461}+\frac{2122346689 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +1}}{61845461}+\frac{2122346689 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{61845461}+\frac{2122346689 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{61845461}+\frac{2122346689 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{61845461}+\frac{2122346689 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{61845461}+\frac{2122346689 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{61845461}+\frac{2122346689 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +1}}{61845461}+\frac{62557958 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{61845461}+\frac{62557958 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{61845461}+\frac{62557958 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{61845461}+\frac{62557958 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{61845461}+\frac{62557958 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{61845461}+\frac{62557958 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =6\right)^{-n -1}}{61845461}+\frac{62557958 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =7\right)^{-n -1}}{61845461}+\frac{62557958 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =8\right)^{-n -1}}{61845461}-\frac{1681531585 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{185536383}-\frac{1681531585 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{185536383}-\frac{1681531585 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{185536383}-\frac{1681531585 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{185536383}-\frac{1681531585 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{185536383}-\frac{1681531585 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =6\right)^{-n}}{185536383}-\frac{1681531585 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =7\right)^{-n}}{185536383}-\frac{1681531585 \mathit{RootOf}\! \left(5 \textit{\_Z}^{8}-44 \textit{\_Z}^{7}+128 \textit{\_Z}^{6}-208 \textit{\_Z}^{5}+209 \textit{\_Z}^{4}-132 \textit{\_Z}^{3}+51 \textit{\_Z}^{2}-11 \textit{\_Z} +1, \mathit{index} =8\right)^{-n}}{185536383}+\left(\left\{\begin{array}{cc}\frac{1}{5} & n =0 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 76
a(6) = 273
a(7) = 964
a(8) = 3356
a(n+8) = -5*a(n)+44*a(n+1)-128*a(n+2)+208*a(n+3)-209*a(n+4)+132*a(n+5)-51*a(n+6)+11*a(n+7), n >= 9
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(3\right) = 6
a \! \left(4\right) = 21
a \! \left(5\right) = 76
a \! \left(6\right) = 273
a \! \left(7\right) = 964
a \! \left(8\right) = 3356
a \! \left(n +8\right) = -5 a \! \left(n \right)+44 a \! \left(n +1\right)-128 a \! \left(n +2\right)+208 a \! \left(n +3\right)-209 a \! \left(n +4\right)+132 a \! \left(n +5\right)-51 a \! \left(n +6\right)+11 a \! \left(n +7\right), \quad n \geq 9
Specification 1
Strategy pack name: insertion_point_placements
Tree: http://permpal.com/tree/380/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[3,x]
F[3,x] = F[16,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[32,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[16,x]*F[8,x]
F[8,x] = F[11,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[6,x]
F[11,x] = F[12,x]*F[16,x]
F[12,x] = F[11,x]+F[13,x]+F[17,x]
F[13,x] = F[1,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[13,x]*F[16,x]
F[16,x] = x
F[17,x] = F[18,x]
F[18,x] = F[13,x]*F[16,x]*F[19,x]
F[19,x] = F[20,x]+F[27,x]
F[20,x] = F[13,x]+F[21,x]
F[21,x] = F[14,x]+F[22,x]
F[22,x] = F[23,x]+F[24,x]+F[26,x]
F[23,x] = 0
F[24,x] = F[16,x]*F[25,x]
F[25,x] = F[14,x]+F[22,x]
F[26,x] = F[16,x]*F[21,x]
F[27,x] = F[14,x]+F[28,x]
F[28,x] = F[23,x]+F[29,x]+F[30,x]
F[29,x] = F[16,x]*F[27,x]
F[30,x] = F[16,x]*F[31,x]
F[31,x] = F[14,x]+F[28,x]
F[32,x] = F[33,x]
F[33,x] = F[16,x]*F[34,x]
F[34,x] = F[35,x]+F[94,x]
F[35,x] = F[36,x]+F[44,x]
F[36,x] = F[14,x]+F[37,x]
F[37,x] = F[2,x]+F[38,x]
F[38,x] = F[39,x]
F[39,x] = F[16,x]*F[40,x]
F[40,x] = F[41,x]+F[42,x]
F[41,x] = F[13,x]*F[2,x]
F[42,x] = F[21,x]+F[43,x]
F[43,x] = F[44,x]+F[65,x]
F[44,x] = F[45,x]
F[45,x] = F[16,x]*F[46,x]
F[46,x] = F[36,x]+F[47,x]
F[47,x] = F[48,x]+F[49,x]
F[48,x] = F[14,x]^2
F[49,x] = F[44,x]+F[50,x]
F[50,x] = F[51,x]
F[51,x] = F[16,x]*F[52,x]
F[52,x] = F[53,x]+F[55,x]
F[53,x] = F[54,x]
F[54,x] = F[14,x]*F[27,x]
F[55,x] = F[56,x]+F[61,x]
F[56,x] = F[38,x]+F[57,x]
F[57,x] = F[58,x]
F[58,x] = F[13,x]*F[14,x]*F[16,x]*F[59,x]
F[59,x] = F[2,x]+F[60,x]
F[60,x] = F[14,x]+F[44,x]
F[61,x] = F[62,x]+F[63,x]
F[62,x] = F[14,x]*F[38,x]
F[63,x] = F[64,x]
F[64,x] = F[13,x]*F[16,x]*F[28,x]*F[59,x]
F[65,x] = F[66,x]
F[66,x] = F[16,x]*F[67,x]
F[67,x] = F[68,x]+F[71,x]
F[68,x] = F[36,x]*F[69,x]
F[69,x] = F[14,x]+F[70,x]
F[70,x] = F[13,x]*F[14,x]
F[71,x] = F[72,x]+F[75,x]
F[72,x] = F[14,x]*F[73,x]
F[73,x] = F[48,x]+F[74,x]
F[74,x] = F[13,x]*F[22,x]
F[75,x] = F[76,x]+F[85,x]
F[76,x] = F[77,x]+F[78,x]
F[77,x] = F[14,x]*F[44,x]
F[78,x] = F[13,x]*F[79,x]
F[79,x] = F[80,x]
F[80,x] = F[16,x]*F[81,x]
F[81,x] = F[82,x]+F[83,x]
F[82,x] = F[44,x]+F[79,x]
F[83,x] = F[84,x]
F[84,x] = F[14,x]*F[2,x]*F[20,x]
F[85,x] = F[86,x]+F[87,x]
F[86,x] = F[14,x]*F[50,x]
F[87,x] = F[13,x]*F[88,x]
F[88,x] = F[89,x]
F[89,x] = F[16,x]*F[90,x]
F[90,x] = F[91,x]+F[92,x]
F[91,x] = F[13,x]*F[50,x]
F[92,x] = F[93,x]
F[93,x] = F[13,x]*F[21,x]*F[38,x]
F[94,x] = F[95,x]+F[97,x]
F[95,x] = F[48,x]+F[96,x]
F[96,x] = F[32,x]+F[50,x]
F[97,x] = F[98,x]
F[98,x] = F[16,x]*F[99,x]
F[99,x] = F[100,x]+F[103,x]
F[100,x] = F[101,x]+F[102,x]
F[101,x] = F[13,x]*F[14,x]*F[36,x]
F[102,x] = F[13,x]*F[44,x]
F[103,x] = F[104,x]+F[106,x]
F[104,x] = F[105,x]
F[105,x] = F[14,x]*F[21,x]*F[36,x]
F[106,x] = F[107,x]+F[97,x]
F[107,x] = F[108,x]*F[14,x]
F[108,x] = F[109,x]
F[109,x] = F[110,x]*F[16,x]
F[110,x] = F[111,x]+F[112,x]
F[111,x] = F[2,x]*F[27,x]
F[112,x] = F[113,x]*F[44,x]
F[113,x] = F[13,x]+F[31,x]
System of equations in latex syntax:
F_{0}\! \left(x \right) = F_{1}\! \left(x \right)+F_{2}\! \left(x \right)
F_{1}\! \left(x \right) = 1
F_{2}\! \left(x \right) = F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{0}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{32}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{16}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{11}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{6}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{12}\! \left(x \right) = F_{11}\! \left(x \right)+F_{13}\! \left(x \right)+F_{17}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{13}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = x
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{13}\! \left(x \right) F_{16}\! \left(x \right) F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{27}\! \left(x \right)
F_{20}\! \left(x \right) = F_{13}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{14}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{26}\! \left(x \right)
F_{23}\! \left(x \right) = 0
F_{24}\! \left(x \right) = F_{16}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{14}\! \left(x \right)+F_{22}\! \left(x \right)
F_{26}\! \left(x \right) = F_{16}\! \left(x \right) F_{21}\! \left(x \right)
F_{27}\! \left(x \right) = F_{14}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{23}\! \left(x \right)+F_{29}\! \left(x \right)+F_{30}\! \left(x \right)
F_{29}\! \left(x \right) = F_{16}\! \left(x \right) F_{27}\! \left(x \right)
F_{30}\! \left(x \right) = F_{16}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{14}\! \left(x \right)+F_{28}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{16}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{94}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{44}\! \left(x \right)
F_{36}\! \left(x \right) = F_{14}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{2}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{16}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{42}\! \left(x \right)
F_{41}\! \left(x \right) = F_{13}\! \left(x \right) F_{2}\! \left(x \right)
F_{42}\! \left(x \right) = F_{21}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{65}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{16}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{36}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{49}\! \left(x \right)
F_{48}\! \left(x \right) = F_{14} \left(x \right)^{2}
F_{49}\! \left(x \right) = F_{44}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{16}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{55}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{14}\! \left(x \right) F_{27}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{61}\! \left(x \right)
F_{56}\! \left(x \right) = F_{38}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{13}\! \left(x \right) F_{14}\! \left(x \right) F_{16}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{2}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{14}\! \left(x \right)+F_{44}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{63}\! \left(x \right)
F_{62}\! \left(x \right) = F_{14}\! \left(x \right) F_{38}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{13}\! \left(x \right) F_{16}\! \left(x \right) F_{28}\! \left(x \right) F_{59}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{16}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)+F_{71}\! \left(x \right)
F_{68}\! \left(x \right) = F_{36}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{14}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{13}\! \left(x \right) F_{14}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)+F_{75}\! \left(x \right)
F_{72}\! \left(x \right) = F_{14}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{48}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{13}\! \left(x \right) F_{22}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)+F_{85}\! \left(x \right)
F_{76}\! \left(x \right) = F_{77}\! \left(x \right)+F_{78}\! \left(x \right)
F_{77}\! \left(x \right) = F_{14}\! \left(x \right) F_{44}\! \left(x \right)
F_{78}\! \left(x \right) = F_{13}\! \left(x \right) F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{16}\! \left(x \right) F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{82}\! \left(x \right)+F_{83}\! \left(x \right)
F_{82}\! \left(x \right) = F_{44}\! \left(x \right)+F_{79}\! \left(x \right)
F_{83}\! \left(x \right) = F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{14}\! \left(x \right) F_{2}\! \left(x \right) F_{20}\! \left(x \right)
F_{85}\! \left(x \right) = F_{86}\! \left(x \right)+F_{87}\! \left(x \right)
F_{86}\! \left(x \right) = F_{14}\! \left(x \right) F_{50}\! \left(x \right)
F_{87}\! \left(x \right) = F_{13}\! \left(x \right) F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = F_{16}\! \left(x \right) F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = F_{91}\! \left(x \right)+F_{92}\! \left(x \right)
F_{91}\! \left(x \right) = F_{13}\! \left(x \right) F_{50}\! \left(x \right)
F_{92}\! \left(x \right) = F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{13}\! \left(x \right) F_{21}\! \left(x \right) F_{38}\! \left(x \right)
F_{94}\! \left(x \right) = F_{95}\! \left(x \right)+F_{97}\! \left(x \right)
F_{95}\! \left(x \right) = F_{48}\! \left(x \right)+F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = F_{32}\! \left(x \right)+F_{50}\! \left(x \right)
F_{97}\! \left(x \right) = F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{16}\! \left(x \right) F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{100}\! \left(x \right)+F_{103}\! \left(x \right)
F_{100}\! \left(x \right) = F_{101}\! \left(x \right)+F_{102}\! \left(x \right)
F_{101}\! \left(x \right) = F_{13}\! \left(x \right) F_{14}\! \left(x \right) F_{36}\! \left(x \right)
F_{102}\! \left(x \right) = F_{13}\! \left(x \right) F_{44}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right)+F_{106}\! \left(x \right)
F_{104}\! \left(x \right) = F_{105}\! \left(x \right)
F_{105}\! \left(x \right) = F_{14}\! \left(x \right) F_{21}\! \left(x \right) F_{36}\! \left(x \right)
F_{106}\! \left(x \right) = F_{107}\! \left(x \right)+F_{97}\! \left(x \right)
F_{107}\! \left(x \right) = F_{108}\! \left(x \right) F_{14}\! \left(x \right)
F_{108}\! \left(x \right) = F_{109}\! \left(x \right)
F_{109}\! \left(x \right) = F_{110}\! \left(x \right) F_{16}\! \left(x \right)
F_{110}\! \left(x \right) = F_{111}\! \left(x \right)+F_{112}\! \left(x \right)
F_{111}\! \left(x \right) = F_{2}\! \left(x \right) F_{27}\! \left(x \right)
F_{112}\! \left(x \right) = F_{113}\! \left(x \right) F_{44}\! \left(x \right)
F_{113}\! \left(x \right) = F_{13}\! \left(x \right)+F_{31}\! \left(x \right)
System of equations in sympy syntax:
Eq(F_0(x), F_1(x) + F_2(x))
Eq(F_1(x), 1)
Eq(F_2(x), F_3(x))
Eq(F_3(x), F_16(x)*F_4(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_32(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_16(x)*F_8(x))
Eq(F_8(x), F_11(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_6(x))
Eq(F_11(x), F_12(x)*F_16(x))
Eq(F_12(x), F_11(x) + F_13(x) + F_17(x))
Eq(F_13(x), F_1(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_13(x)*F_16(x))
Eq(F_16(x), x)
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_13(x)*F_16(x)*F_19(x))
Eq(F_19(x), F_20(x) + F_27(x))
Eq(F_20(x), F_13(x) + F_21(x))
Eq(F_21(x), F_14(x) + F_22(x))
Eq(F_22(x), F_23(x) + F_24(x) + F_26(x))
Eq(F_23(x), 0)
Eq(F_24(x), F_16(x)*F_25(x))
Eq(F_25(x), F_14(x) + F_22(x))
Eq(F_26(x), F_16(x)*F_21(x))
Eq(F_27(x), F_14(x) + F_28(x))
Eq(F_28(x), F_23(x) + F_29(x) + F_30(x))
Eq(F_29(x), F_16(x)*F_27(x))
Eq(F_30(x), F_16(x)*F_31(x))
Eq(F_31(x), F_14(x) + F_28(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_16(x)*F_34(x))
Eq(F_34(x), F_35(x) + F_94(x))
Eq(F_35(x), F_36(x) + F_44(x))
Eq(F_36(x), F_14(x) + F_37(x))
Eq(F_37(x), F_2(x) + F_38(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_16(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_42(x))
Eq(F_41(x), F_13(x)*F_2(x))
Eq(F_42(x), F_21(x) + F_43(x))
Eq(F_43(x), F_44(x) + F_65(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_16(x)*F_46(x))
Eq(F_46(x), F_36(x) + F_47(x))
Eq(F_47(x), F_48(x) + F_49(x))
Eq(F_48(x), F_14(x)**2)
Eq(F_49(x), F_44(x) + F_50(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_16(x)*F_52(x))
Eq(F_52(x), F_53(x) + F_55(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_14(x)*F_27(x))
Eq(F_55(x), F_56(x) + F_61(x))
Eq(F_56(x), F_38(x) + F_57(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_13(x)*F_14(x)*F_16(x)*F_59(x))
Eq(F_59(x), F_2(x) + F_60(x))
Eq(F_60(x), F_14(x) + F_44(x))
Eq(F_61(x), F_62(x) + F_63(x))
Eq(F_62(x), F_14(x)*F_38(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_13(x)*F_16(x)*F_28(x)*F_59(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_16(x)*F_67(x))
Eq(F_67(x), F_68(x) + F_71(x))
Eq(F_68(x), F_36(x)*F_69(x))
Eq(F_69(x), F_14(x) + F_70(x))
Eq(F_70(x), F_13(x)*F_14(x))
Eq(F_71(x), F_72(x) + F_75(x))
Eq(F_72(x), F_14(x)*F_73(x))
Eq(F_73(x), F_48(x) + F_74(x))
Eq(F_74(x), F_13(x)*F_22(x))
Eq(F_75(x), F_76(x) + F_85(x))
Eq(F_76(x), F_77(x) + F_78(x))
Eq(F_77(x), F_14(x)*F_44(x))
Eq(F_78(x), F_13(x)*F_79(x))
Eq(F_79(x), F_80(x))
Eq(F_80(x), F_16(x)*F_81(x))
Eq(F_81(x), F_82(x) + F_83(x))
Eq(F_82(x), F_44(x) + F_79(x))
Eq(F_83(x), F_84(x))
Eq(F_84(x), F_14(x)*F_2(x)*F_20(x))
Eq(F_85(x), F_86(x) + F_87(x))
Eq(F_86(x), F_14(x)*F_50(x))
Eq(F_87(x), F_13(x)*F_88(x))
Eq(F_88(x), F_89(x))
Eq(F_89(x), F_16(x)*F_90(x))
Eq(F_90(x), F_91(x) + F_92(x))
Eq(F_91(x), F_13(x)*F_50(x))
Eq(F_92(x), F_93(x))
Eq(F_93(x), F_13(x)*F_21(x)*F_38(x))
Eq(F_94(x), F_95(x) + F_97(x))
Eq(F_95(x), F_48(x) + F_96(x))
Eq(F_96(x), F_32(x) + F_50(x))
Eq(F_97(x), F_98(x))
Eq(F_98(x), F_16(x)*F_99(x))
Eq(F_99(x), F_100(x) + F_103(x))
Eq(F_100(x), F_101(x) + F_102(x))
Eq(F_101(x), F_13(x)*F_14(x)*F_36(x))
Eq(F_102(x), F_13(x)*F_44(x))
Eq(F_103(x), F_104(x) + F_106(x))
Eq(F_104(x), F_105(x))
Eq(F_105(x), F_14(x)*F_21(x)*F_36(x))
Eq(F_106(x), F_107(x) + F_97(x))
Eq(F_107(x), F_108(x)*F_14(x))
Eq(F_108(x), F_109(x))
Eq(F_109(x), F_110(x)*F_16(x))
Eq(F_110(x), F_111(x) + F_112(x))
Eq(F_111(x), F_2(x)*F_27(x))
Eq(F_112(x), F_113(x)*F_44(x))
Eq(F_113(x), F_13(x) + F_31(x))
Pack JSON:
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Specification JSON:
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