0231_0321_1302_2013
Counting sequence:
1, 1, 2, 6, 20, 65, 206, 645, 2012, 6273, 19563, 61024, 190377, 593940, 1852986, 5780966, 18035478, 56267089, 175542029, 547655902, 1708576564, 5330416293, 16629830192, 51881736376, 161860616641, 504972675171, 1575413512587, 4914974329221, 15333734580415, 47838177868990, 149245524619711, 465616116902725, 1452628940614227, 4531911079768758, 14138654036622808, 44109766155761154, 137613627526078117, 429326929868854220, 1339413952122484601, 4178703012383577316, 13036715675563452113, 40671939379711303479, 126888297181139199534, 395866049347051899687, 1235022712945162339044, 3853023273923725150858, 12020660182026212175632, 37502049933013436439126, 116998877589198145197181, 365013042795877595401591, 1138767517744169969145472, 3552726361601301774210386, 11083794017430334727193635, 34579215317177628169537890, 107880219541372165422854516, 336564657744366867688363555, 1050014259557021432670626717, 3275833988815580507415034393, 10219945324177419814122046753, 31884180573796461386670745759, 99472251427556785754397394582, 310333482811815098089817334925, 968178252447106633449146208941, 3020522052658913340839421328741, 9423423266881582674444659229298, 29399191437332913297120022747394, 91719583498552625955814238201064, 286146713084948204770794705909997, 892720379727975634062683827514574, 2785108617148682858705094297071573, 8688980542462201808013577264140572, 27107877374125499714906052897364501, 84571142971209137557905079478172371, 263845011719123185292571263668360636, 823143542387304183459560980917899657, 2568042833022074830545319825876079376, 8011778812124907199788290160019161430, 24995143737098961968530132409575499390, 77979837572742169117395141103539043746, 243281460264049678884260727801558466385, 758989384313577836074159309005907767357, 2367894721099840952270428848388378540678, 7387356827504694321752071412102716319148, 23047072325720675639740959363312233002089, 71902245307733146796515641367217543466772, 224320590799021191538114403916981084590680, 699835272752045504725419122080136371789881, 2183345751914215280207571554397988646899991, 6811601040993567060527557452551054739674283, 21250829696114774091640756914676506911155709, 66298328404066616693046320908962408977747619, 206837493501586751061018702486445501651681082, 645291514097884679569539080164224286980087779, 2013180159541751971622457402519692109354551313, 6280718506640345813603021443811475353911524887, 19594582617302226190795513725976133459341643126, 61131169553348148781701850188601419219699689836, 190716993770531817277425094917483747876570536806, 594998786684867663143182649930240463821448283017, 1856276932418624147687031141802761133195118488316, 5791211893100036619637760630905800367835223807425
Generating function in Maple syntax:
-(x^2+2*x-1)*(x-1)^2/(x^5-2*x^4-3*x^3+7*x^2-5*x+1)
Generating function in latex syntax:
-\frac{\left(x^{2}+2 x -1\right) \left(x -1\right)^{2}}{x^{5}-2 x^{4}-3 x^{3}+7 x^{2}-5 x +1}
Generating function in sympy syntax:
(x - 1)**2*(-x**2 - 2*x + 1)/(x**5 - 2*x**4 - 3*x**3 + 7*x**2 - 5*x + 1)
Implicit equation for the generating function in Maple syntax:
(x^5-2*x^4-3*x^3+7*x^2-5*x+1)*F(x)+(x^2+2*x-1)*(x-1)^2 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{5}-2 x^{4}-3 x^{3}+7 x^{2}-5 x +1\right) F \! \left(x \right)+\left(x^{2}+2 x -1\right) \left(x -1\right)^{2} = 0
Explicit closed form in Maple syntax:
2390/25679*RootOf(_Z^5-2*_Z^4-3*_Z^3+7*_Z^2-5*_Z+1,index = 1)^(-n+3)+2390/25679*RootOf(_Z^5-2*_Z^4-3*_Z^3+7*_Z^2-5*_Z+1,index = 2)^(-n+3)+2390/25679*RootOf(_Z^5-2*_Z^4-3*_Z^3+7*_Z^2-5*_Z+1,index = 3)^(-n+3)+2390/25679*RootOf(_Z^5-2*_Z^4-3*_Z^3+7*_Z^2-5*_Z+1,index = 4)^(-n+3)+2390/25679*RootOf(_Z^5-2*_Z^4-3*_Z^3+7*_Z^2-5*_Z+1,index = 5)^(-n+3)-3145/25679*RootOf(_Z^5-2*_Z^4-3*_Z^3+7*_Z^2-5*_Z+1,index = 1)^(-n+2)-3145/25679*RootOf(_Z^5-2*_Z^4-3*_Z^3+7*_Z^2-5*_Z+1,index = 2)^(-n+2)-3145/25679*RootOf(_Z^5-2*_Z^4-3*_Z^3+7*_Z^2-5*_Z+1,index = 3)^(-n+2)-3145/25679*RootOf(_Z^5-2*_Z^4-3*_Z^3+7*_Z^2-5*_Z+1,index = 4)^(-n+2)-3145/25679*RootOf(_Z^5-2*_Z^4-3*_Z^3+7*_Z^2-5*_Z+1,index = 5)^(-n+2)-8838/25679*RootOf(_Z^5-2*_Z^4-3*_Z^3+7*_Z^2-5*_Z+1,index = 1)^(-n+1)-8838/25679*RootOf(_Z^5-2*_Z^4-3*_Z^3+7*_Z^2-5*_Z+1,index = 2)^(-n+1)-8838/25679*RootOf(_Z^5-2*_Z^4-3*_Z^3+7*_Z^2-5*_Z+1,index = 3)^(-n+1)-8838/25679*RootOf(_Z^5-2*_Z^4-3*_Z^3+7*_Z^2-5*_Z+1,index = 4)^(-n+1)-8838/25679*RootOf(_Z^5-2*_Z^4-3*_Z^3+7*_Z^2-5*_Z+1,index = 5)^(-n+1)-1766/25679*RootOf(_Z^5-2*_Z^4-3*_Z^3+7*_Z^2-5*_Z+1,index = 1)^(-n-1)-1766/25679*RootOf(_Z^5-2*_Z^4-3*_Z^3+7*_Z^2-5*_Z+1,index = 2)^(-n-1)-1766/25679*RootOf(_Z^5-2*_Z^4-3*_Z^3+7*_Z^2-5*_Z+1,index = 3)^(-n-1)-1766/25679*RootOf(_Z^5-2*_Z^4-3*_Z^3+7*_Z^2-5*_Z+1,index = 4)^(-n-1)-1766/25679*RootOf(_Z^5-2*_Z^4-3*_Z^3+7*_Z^2-5*_Z+1,index = 5)^(-n-1)+14337/25679*RootOf(_Z^5-2*_Z^4-3*_Z^3+7*_Z^2-5*_Z+1,index = 1)^(-n)+14337/25679*RootOf(_Z^5-2*_Z^4-3*_Z^3+7*_Z^2-5*_Z+1,index = 2)^(-n)+14337/25679*RootOf(_Z^5-2*_Z^4-3*_Z^3+7*_Z^2-5*_Z+1,index = 3)^(-n)+14337/25679*RootOf(_Z^5-2*_Z^4-3*_Z^3+7*_Z^2-5*_Z+1,index = 4)^(-n)+14337/25679*RootOf(_Z^5-2*_Z^4-3*_Z^3+7*_Z^2-5*_Z+1,index = 5)^(-n)
Explicit closed form in latex syntax:
\frac{2390 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{25679}+\frac{2390 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{25679}+\frac{2390 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{25679}+\frac{2390 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{25679}+\frac{2390 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{25679}-\frac{3145 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{25679}-\frac{3145 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{25679}-\frac{3145 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{25679}-\frac{3145 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{25679}-\frac{3145 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{25679}-\frac{8838 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{25679}-\frac{8838 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{25679}-\frac{8838 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{25679}-\frac{8838 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{25679}-\frac{8838 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{25679}-\frac{1766 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{25679}-\frac{1766 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{25679}-\frac{1766 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{25679}-\frac{1766 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{25679}-\frac{1766 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{25679}+\frac{14337 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{25679}+\frac{14337 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{25679}+\frac{14337 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{25679}+\frac{14337 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{25679}+\frac{14337 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-2 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{25679}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(n+5) = -a(n)+2*a(n+1)+3*a(n+2)-7*a(n+3)+5*a(n+4), n >= 5
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) = 20
a \! \left(n +5\right) = -a \! \left(n \right)+2 a \! \left(n +1\right)+3 a \! \left(n +2\right)-7 a \! \left(n +3\right)+5 a \! \left(n +4\right), \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15826/
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[17,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[6,x]
F[10,x] = F[11,x]+F[14,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[1,x]+F[11,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[7,x]
F[17,x] = F[18,x]+F[2,x]
F[18,x] = F[19,x]+F[20,x]+F[37,x]
F[19,x] = 0
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[26,x]
F[22,x] = F[23,x]+F[7,x]
F[23,x] = F[24,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[11,x]
F[26,x] = F[18,x]+F[27,x]
F[27,x] = 2*F[19,x]+F[28,x]+F[32,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[30,x]+F[31,x]
F[30,x] = F[23,x]
F[31,x] = F[27,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[2,x]+F[34,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[17,x]+F[39,x]
F[39,x] = F[34,x]+F[40,x]
F[40,x] = F[41,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[18,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_{17}\! \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_{6}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{7}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{2}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{37}\! \left(x \right)
F_{19}\! \left(x \right) = 0
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{26}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{7}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{11}\! \left(x \right)
F_{26}\! \left(x \right) = F_{18}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{28}\! \left(x \right)+F_{32}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{31}\! \left(x \right)
F_{30}\! \left(x \right) = F_{23}\! \left(x \right)
F_{31}\! \left(x \right) = F_{27}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{2}\! \left(x \right)+F_{34}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{17}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{34}\! \left(x \right)+F_{40}\! \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_{18}\! \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_17(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_6(x))
Eq(F_10(x), F_11(x) + F_14(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_1(x) + F_11(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_7(x))
Eq(F_17(x), F_18(x) + F_2(x))
Eq(F_18(x), F_19(x) + F_20(x) + F_37(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_26(x))
Eq(F_22(x), F_23(x) + F_7(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_11(x))
Eq(F_26(x), F_18(x) + F_27(x))
Eq(F_27(x), 2*F_19(x) + F_28(x) + F_32(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_30(x) + F_31(x))
Eq(F_30(x), F_23(x))
Eq(F_31(x), F_27(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_2(x) + F_34(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_17(x) + F_39(x))
Eq(F_39(x), F_34(x) + F_40(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_18(x))
Pack JSON:
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Specification JSON:
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