0132_1032_1302_2031_2103
Counting sequence:
1, 1, 2, 6, 19, 58, 173, 514, 1533, 4591, 13782, 41418, 124531, 374529, 1126620, 3389456, 10198198, 30686180, 92337516, 277858305, 836131781, 2516112090, 7571601117, 22784896025, 68565773134, 206332830651, 620911486808, 1868492266215, 5622805673877, 16920568290923, 50918649865222, 153228254543611, 461106086525158, 1387595463808862, 4175657777446034, 12565707072552863, 37813682090925351, 113791810800128406, 342430981005501938, 1030469382695403539, 3100966936899348550, 9331665854764483972, 28081559542316489488, 84505167555440180825, 254299385848835167472, 765257078633596987888, 2302869881106188270218, 6929971427506834128418, 20854197791165340482034, 62756040204926271861110, 188850255555322452099659, 568302571468768248886502, 1710179378843291007018768, 5146402030652988452125703, 15486944930354697614622732, 46604494139331622999034690, 140245793069749348205824863, 422038321346689973752528559, 1270029858197729625199965588, 3821870572243939802236630071, 11501063992081682621403717056, 34609877663206292922968503964, 104150679683799623974089898357, 313418157213819866205182333879, 943161787993484865129046077063, 2838234281762465155611577736941, 8541030754977345724254640029814, 25702320215852620749196621551925, 77345379431312112746196178344753, 232753606255512270082522774000251, 700419877997450243585531758893694, 2107756839459689604274105882893764, 6342822403885680753982631466143332, 19087304234556397455067672439027674, 57438969553889003222028903223453003, 172849721619647301113065918063892713, 520152546190075162205235370781874939, 1565282655782216273101714312279682881, 4710367776604719986274973430508230685, 14174797445633444776164003284015004977, 42655879997880991563906012523323584465, 128363322676905638389461862290582367074, 386280686495599890287656636766713590200, 1162425260174083958736632852666700912384, 3498058620919890664222963286646289898049, 10526624407284001129744208572576681156731, 31677519853251456785071217262159680568135, 95326405239535529872065370659095509480786, 286863478516908662675038127673867171048317, 863251426507078809847282563952711847613691, 2597761936162890834790002480326852271171858, 7817383058701998622469779753572255933022792, 23524664456646755414297583610652160193730856, 70792211874763569882509401652424165342861131, 213033315368093841247916773796301760732787876, 641076076800761740469015361613634309755605847, 1929174953392330896197094068536739326349156968, 5805420191889275413595223117039181941421502540, 17470112570729424374445607345399522001967975383, 52572393236988850378217035210351031810897795544, 158204849538024043777731116050312158280909466855
Generating function in Maple syntax:
-(3*x^3-5*x^2+4*x-1)*(x-1)^3/(x^7-12*x^6+34*x^5-52*x^4+47*x^3-26*x^2+8*x-1)
Generating function in latex syntax:
-\frac{\left(3 x^{3}-5 x^{2}+4 x -1\right) \left(x -1\right)^{3}}{x^{7}-12 x^{6}+34 x^{5}-52 x^{4}+47 x^{3}-26 x^{2}+8 x -1}
Generating function in sympy syntax:
(x - 1)**3*(-3*x**3 + 5*x**2 - 4*x + 1)/(x**7 - 12*x**6 + 34*x**5 - 52*x**4 + 47*x**3 - 26*x**2 + 8*x - 1)
Implicit equation for the generating function in Maple syntax:
(x^7-12*x^6+34*x^5-52*x^4+47*x^3-26*x^2+8*x-1)*F(x)+(3*x^3-5*x^2+4*x-1)*(x-1)^3 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{7}-12 x^{6}+34 x^{5}-52 x^{4}+47 x^{3}-26 x^{2}+8 x -1\right) F \! \left(x \right)+\left(3 x^{3}-5 x^{2}+4 x -1\right) \left(x -1\right)^{3} = 0
Explicit closed form in Maple syntax:
17672/128257*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 1)^(-n+5)+17672/128257*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 2)^(-n+5)+17672/128257*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 3)^(-n+5)+17672/128257*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 4)^(-n+5)+17672/128257*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 5)^(-n+5)+17672/128257*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 6)^(-n+5)+17672/128257*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 7)^(-n+5)-612028/384771*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 1)^(-n+4)-612028/384771*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 2)^(-n+4)-612028/384771*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 3)^(-n+4)-612028/384771*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 4)^(-n+4)-612028/384771*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 5)^(-n+4)-612028/384771*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 6)^(-n+4)-612028/384771*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 7)^(-n+4)+505973/128257*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 1)^(-n+3)+505973/128257*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 2)^(-n+3)+505973/128257*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 3)^(-n+3)+505973/128257*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 4)^(-n+3)+505973/128257*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 5)^(-n+3)+505973/128257*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 6)^(-n+3)+505973/128257*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 7)^(-n+3)-666681/128257*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 1)^(-n+2)-666681/128257*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 2)^(-n+2)-666681/128257*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 3)^(-n+2)-666681/128257*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 4)^(-n+2)-666681/128257*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 5)^(-n+2)-666681/128257*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 6)^(-n+2)-666681/128257*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 7)^(-n+2)+1439041/384771*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 1)^(-n+1)+1439041/384771*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 2)^(-n+1)+1439041/384771*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 3)^(-n+1)+1439041/384771*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 4)^(-n+1)+1439041/384771*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 5)^(-n+1)+1439041/384771*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 6)^(-n+1)+1439041/384771*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 7)^(-n+1)+113239/384771*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 1)^(-n-1)+113239/384771*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 2)^(-n-1)+113239/384771*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 3)^(-n-1)+113239/384771*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 4)^(-n-1)+113239/384771*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 5)^(-n-1)+113239/384771*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 6)^(-n-1)+113239/384771*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 7)^(-n-1)-559823/384771*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 1)^(-n)-559823/384771*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 2)^(-n)-559823/384771*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 3)^(-n)-559823/384771*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 4)^(-n)-559823/384771*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 5)^(-n)-559823/384771*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 6)^(-n)-559823/384771*RootOf(_Z^7-12*_Z^6+34*_Z^5-52*_Z^4+47*_Z^3-26*_Z^2+8*_Z-1,index = 7)^(-n)
Explicit closed form in latex syntax:
\frac{17672 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +5}}{128257}+\frac{17672 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +5}}{128257}+\frac{17672 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +5}}{128257}+\frac{17672 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +5}}{128257}+\frac{17672 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +5}}{128257}+\frac{17672 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +5}}{128257}+\frac{17672 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +5}}{128257}-\frac{612028 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +4}}{384771}-\frac{612028 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +4}}{384771}-\frac{612028 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +4}}{384771}-\frac{612028 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +4}}{384771}-\frac{612028 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +4}}{384771}-\frac{612028 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +4}}{384771}-\frac{612028 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +4}}{384771}+\frac{505973 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +3}}{128257}+\frac{505973 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +3}}{128257}+\frac{505973 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +3}}{128257}+\frac{505973 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +3}}{128257}+\frac{505973 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +3}}{128257}+\frac{505973 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +3}}{128257}+\frac{505973 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +3}}{128257}-\frac{666681 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +2}}{128257}-\frac{666681 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +2}}{128257}-\frac{666681 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +2}}{128257}-\frac{666681 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +2}}{128257}-\frac{666681 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +2}}{128257}-\frac{666681 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +2}}{128257}-\frac{666681 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +2}}{128257}+\frac{1439041 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +1}}{384771}+\frac{1439041 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +1}}{384771}+\frac{1439041 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +1}}{384771}+\frac{1439041 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +1}}{384771}+\frac{1439041 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +1}}{384771}+\frac{1439041 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +1}}{384771}+\frac{1439041 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +1}}{384771}+\frac{113239 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =1\right)^{-n -1}}{384771}+\frac{113239 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =2\right)^{-n -1}}{384771}+\frac{113239 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =3\right)^{-n -1}}{384771}+\frac{113239 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =4\right)^{-n -1}}{384771}+\frac{113239 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =5\right)^{-n -1}}{384771}+\frac{113239 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =6\right)^{-n -1}}{384771}+\frac{113239 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =7\right)^{-n -1}}{384771}-\frac{559823 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =1\right)^{-n}}{384771}-\frac{559823 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =2\right)^{-n}}{384771}-\frac{559823 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =3\right)^{-n}}{384771}-\frac{559823 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =4\right)^{-n}}{384771}-\frac{559823 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =5\right)^{-n}}{384771}-\frac{559823 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =6\right)^{-n}}{384771}-\frac{559823 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-12 \textit{\_Z}^{6}+34 \textit{\_Z}^{5}-52 \textit{\_Z}^{4}+47 \textit{\_Z}^{3}-26 \textit{\_Z}^{2}+8 \textit{\_Z} -1, \mathit{index} =7\right)^{-n}}{384771}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 58
a(6) = 173
a(n+7) = a(n)-12*a(n+1)+34*a(n+2)-52*a(n+3)+47*a(n+4)-26*a(n+5)+8*a(n+6), n >= 7
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) = 19
a \! \left(5\right) = 58
a \! \left(6\right) = 173
a \! \left(n +7\right) = a \! \left(n \right)-12 a \! \left(n +1\right)+34 a \! \left(n +2\right)-52 a \! \left(n +3\right)+47 a \! \left(n +4\right)-26 a \! \left(n +5\right)+8 a \! \left(n +6\right), \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19774/
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[4,x]*F[5,x]
F[4,x] = x
F[5,x] = F[28,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[14,x]+F[7,x]
F[14,x] = F[15,x]+F[16,x]+F[27,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[18,x]+F[21,x]
F[18,x] = F[11,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[18,x]*F[4,x]
F[21,x] = F[22,x]+F[25,x]
F[22,x] = F[23,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[22,x]+F[7,x]
F[25,x] = F[26,x]
F[26,x] = F[21,x]*F[4,x]
F[27,x] = F[13,x]*F[4,x]
F[28,x] = F[2,x]+F[29,x]
F[29,x] = F[15,x]+F[30,x]+F[46,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[32,x]+F[36,x]
F[32,x] = F[11,x]+F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[18,x]
F[36,x] = F[37,x]+F[40,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[2,x]+F[37,x]
F[40,x] = F[41,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[43,x]
F[43,x] = F[37,x]+F[44,x]
F[44,x] = F[45,x]
F[45,x] = F[4,x]*F[43,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]+F[51,x]
F[48,x] = F[2,x]+F[49,x]
F[49,x] = F[15,x]+F[30,x]+F[50,x]
F[50,x] = F[4,x]*F[48,x]
F[51,x] = F[52,x]+F[69,x]
F[52,x] = F[53,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[39,x]+F[55,x]
F[55,x] = F[52,x]+F[56,x]
F[56,x] = 2*F[15,x]+F[57,x]+F[68,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[59,x]+F[62,x]
F[59,x] = F[37,x]+F[60,x]
F[60,x] = F[61,x]
F[61,x] = F[4,x]*F[59,x]
F[62,x] = F[63,x]+F[66,x]
F[63,x] = F[64,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[52,x]+F[63,x]
F[66,x] = F[67,x]
F[67,x] = F[4,x]*F[62,x]
F[68,x] = F[4,x]*F[55,x]
F[69,x] = 2*F[15,x]+F[57,x]+F[70,x]
F[70,x] = F[4,x]*F[51,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_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{28}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{13}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{27}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = F_{11}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{25}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{22}\! \left(x \right)+F_{7}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{2}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{15}\! \left(x \right)+F_{30}\! \left(x \right)+F_{46}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{36}\! \left(x \right)
F_{32}\! \left(x \right) = F_{11}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{18}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{40}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{2}\! \left(x \right)+F_{37}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{37}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{51}\! \left(x \right)
F_{48}\! \left(x \right) = F_{2}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{15}\! \left(x \right)+F_{30}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{69}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{39}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{52}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{57}\! \left(x \right)+F_{68}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{62}\! \left(x \right)
F_{59}\! \left(x \right) = F_{37}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)+F_{66}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{52}\! \left(x \right)+F_{63}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{69}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{57}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \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_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_28(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_7(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_27(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_21(x))
Eq(F_18(x), F_11(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_18(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_25(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_22(x) + F_7(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_21(x)*F_4(x))
Eq(F_27(x), F_13(x)*F_4(x))
Eq(F_28(x), F_2(x) + F_29(x))
Eq(F_29(x), F_15(x) + F_30(x) + F_46(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_32(x) + F_36(x))
Eq(F_32(x), F_11(x) + F_33(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_18(x))
Eq(F_36(x), F_37(x) + F_40(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_2(x) + F_37(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_37(x) + F_44(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_4(x)*F_43(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_51(x))
Eq(F_48(x), F_2(x) + F_49(x))
Eq(F_49(x), F_15(x) + F_30(x) + F_50(x))
Eq(F_50(x), F_4(x)*F_48(x))
Eq(F_51(x), F_52(x) + F_69(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_39(x) + F_55(x))
Eq(F_55(x), F_52(x) + F_56(x))
Eq(F_56(x), 2*F_15(x) + F_57(x) + F_68(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_59(x) + F_62(x))
Eq(F_59(x), F_37(x) + F_60(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_4(x)*F_59(x))
Eq(F_62(x), F_63(x) + F_66(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_52(x) + F_63(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), F_4(x)*F_62(x))
Eq(F_68(x), F_4(x)*F_55(x))
Eq(F_69(x), 2*F_15(x) + F_57(x) + F_70(x))
Eq(F_70(x), F_4(x)*F_51(x))
Pack JSON:
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Specification JSON:
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