0231_1230_2031_2103_3012
Counting sequence:
1, 1, 2, 6, 19, 50, 116, 261, 574, 1250, 2724, 5964, 13147, 29199, 65320, 147087, 333113, 758099, 1732323, 3971826, 9131512, 21040745, 48568938, 112274686, 259840036, 601908344, 1395319075, 3236461571, 7510503784, 17435236359, 40486821789, 94037332843, 218457415243, 507570720614, 1179441643044, 2740918024417, 6370111926374, 14805495891734, 34412683904820, 79988898851984, 185931608208675, 432201723449723, 1004679375971536, 2335471608604675, 5429083652511713, 12620666814056543, 29338715618081571, 68202813303923878, 158549682485035064, 368578470758319837, 856832118322444218, 1991876694937709410, 4630521509935152324, 10764601305976686916, 25024560430732243635, 58174858266646084271, 135239794206377009504, 314393746392700013139, 730875625946515145613, 1699077555013100927047, 3949872172082202959963, 9182331537377254175682, 21346317716486598501772, 49624143658047193542133, 115362092888521814198054, 268184246180699674806418, 623452580759080490187164, 1449351130065060032962252, 3369332083569134812550403, 7832745851644454879582223, 18208923194140357181322928, 42330607528001004768968815, 98406717569254755147839393, 228767852609370523717228275, 531820715068258684750909627, 1236333129691301292046367186, 2874125760296892090191024912, 6681531642313134580273953849, 15532676362659869651445651842, 36109091244760117951818763134, 83943452064743386208679806012, 195144848988572264316486572632, 453656732096772133570237454107, 1054623946728369362391217502867, 2451703218267574655483420266160, 5699518481096432045562936404103, 13249772930774083533676613912245, 30801985006629641752409446173659, 71605927562615112838348404965379, 166463585464062309079398779033382, 386980886013699117039337230758284, 899621414197580712661396997024961, 2091366055150214024109673866429886, 4861836221452093215759586109102070, 11302397965279146441423421880264252, 26274887502612902309402161408752704, 61081702788670570235077256256461883, 141997731307558265024052905922300843, 330104675813907267774107828554251304, 767400267552667829842156459737919571, 1783989182222593128694071657672537657
Generating function in Maple syntax:
-(x^11-2*x^9+10*x^8-6*x^7+10*x^5-4*x^4+6*x^3-9*x^2+5*x-1)/(x^3-2*x^2+3*x-1)/(x^3+x^2+x-1)/(x-1)^2
Generating function in latex syntax:
-\frac{x^{11}-2 x^{9}+10 x^{8}-6 x^{7}+10 x^{5}-4 x^{4}+6 x^{3}-9 x^{2}+5 x -1}{\left(x^{3}-2 x^{2}+3 x -1\right) \left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(-x**11 + 2*x**9 - 10*x**8 + 6*x**7 - 10*x**5 + 4*x**4 - 6*x**3 + 9*x**2 - 5*x + 1)/((x - 1)**2*(x**3 - 2*x**2 + 3*x - 1)*(x**3 + x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(x^3-2*x^2+3*x-1)*(x^3+x^2+x-1)*(x-1)^2*F(x)+x^11-2*x^9+10*x^8-6*x^7+10*x^5-4*x^4+6*x^3-9*x^2+5*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{3}-2 x^{2}+3 x -1\right) \left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{11}-2 x^{9}+10 x^{8}-6 x^{7}+10 x^{5}-4 x^{4}+6 x^{3}-9 x^{2}+5 x -1 = 0
Explicit closed form in Maple syntax:
5-5*n+381/253*Sum(_alpha^(-n+4),_alpha = RootOf(_Z^6-_Z^5+2*_Z^4-_Z^3+4*_Z^2-4*_Z+1))-223/253*Sum(_alpha^(3-n),_alpha = RootOf(_Z^6-_Z^5+2*_Z^4-_Z^3+4*_Z^2-4*_Z+1))+68/23*Sum(_alpha^(2-n),_alpha = RootOf(_Z^6-_Z^5+2*_Z^4-_Z^3+4*_Z^2-4*_Z+1))-152/253*Sum(_alpha^(1-n),_alpha = RootOf(_Z^6-_Z^5+2*_Z^4-_Z^3+4*_Z^2-4*_Z+1))+1598/253*Sum(_alpha^(-n),_alpha = RootOf(_Z^6-_Z^5+2*_Z^4-_Z^3+4*_Z^2-4*_Z+1))-672/253*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^6-_Z^5+2*_Z^4-_Z^3+4*_Z^2-4*_Z+1))-piecewise(n = 0,7,n = 1,2,n = 2,3,n = 3,1,0)
Explicit closed form in latex syntax:
5-5 n +\frac{381 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +4}\right)}{253}-\frac{223 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{3-n}\right)}{253}+\frac{68 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{2-n}\right)}{23}-\frac{152 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{1-n}\right)}{253}+\frac{1598 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{253}-\frac{672 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}-\textit{\_Z}^{5}+2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{253}-\left(\left\{\begin{array}{cc}7 & n =0 \\ 2 & n =1 \\ 3 & n =2 \\ 1 & n =3 \\ 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) = 19
a(5) = 50
a(6) = 116
a(7) = 261
a(8) = 574
a(9) = 1250
a(10) = 2724
a(11) = 5964
a(n+6) = -a(n)+a(n+1)-2*a(n+2)+a(n+3)-4*a(n+4)+4*a(n+5)-10*n, n >= 12
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) = 50
a \! \left(6\right) = 116
a \! \left(7\right) = 261
a \! \left(8\right) = 574
a \! \left(9\right) = 1250
a \! \left(10\right) = 2724
a \! \left(11\right) = 5964
a \! \left(n +6\right) = -a \! \left(n \right)+a \! \left(n +1\right)-2 a \! \left(n +2\right)+a \! \left(n +3\right)-4 a \! \left(n +4\right)+4 a \! \left(n +5\right)-10 n, \quad n \geq 12
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/21071/
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[29,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[22,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[14,x]+F[17,x]
F[14,x] = F[1,x]+F[15,x]
F[15,x] = F[16,x]
F[16,x] = F[14,x]*F[4,x]
F[17,x] = F[11,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]
F[21,x] = F[15,x]
F[22,x] = F[23,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]
F[25,x] = F[26,x]+F[7,x]
F[26,x] = F[27,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[7,x]
F[29,x] = F[30,x]+F[43,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[14,x]+F[33,x]
F[33,x] = F[30,x]+F[34,x]
F[34,x] = F[35,x]+F[36,x]+F[42,x]
F[35,x] = 0
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[21,x]+F[38,x]
F[38,x] = F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[30,x]+F[39,x]
F[42,x] = F[33,x]*F[4,x]
F[43,x] = F[35,x]+F[44,x]+F[55,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[25,x]+F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]+F[50,x]
F[49,x] = F[30,x]
F[50,x] = F[51,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[53,x]+F[54,x]
F[53,x] = F[30,x]
F[54,x] = F[51,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[57,x]+F[61,x]
F[57,x] = F[30,x]+F[58,x]
F[58,x] = F[59,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[25,x]
F[61,x] = F[62,x]+F[70,x]
F[62,x] = F[35,x]+F[36,x]+F[63,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[65,x]+F[69,x]
F[65,x] = F[30,x]+F[66,x]
F[66,x] = F[67,x]
F[67,x] = F[4,x]*F[68,x]
F[68,x] = F[21,x]
F[69,x] = F[62,x]
F[70,x] = F[71,x]
F[71,x] = F[4,x]*F[72,x]
F[72,x] = F[26,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_{29}\! \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_{22}\! \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_{14}\! \left(x \right)+F_{17}\! \left(x \right)
F_{14}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{11}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{15}\! \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_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{7}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{7}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{43}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{14}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{30}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{36}\! \left(x \right)+F_{42}\! \left(x \right)
F_{35}\! \left(x \right) = 0
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{21}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{30}\! \left(x \right)+F_{39}\! \left(x \right)
F_{42}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{43}\! \left(x \right) = F_{35}\! \left(x \right)+F_{44}\! \left(x \right)+F_{55}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{25}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{50}\! \left(x \right)
F_{49}\! \left(x \right) = F_{30}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{54}\! \left(x \right)
F_{53}\! \left(x \right) = F_{30}\! \left(x \right)
F_{54}\! \left(x \right) = F_{51}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{61}\! \left(x \right)
F_{57}\! \left(x \right) = F_{30}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{25}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{70}\! \left(x \right)
F_{62}\! \left(x \right) = F_{35}\! \left(x \right)+F_{36}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{69}\! \left(x \right)
F_{65}\! \left(x \right) = F_{30}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{4}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{21}\! \left(x \right)
F_{69}\! \left(x \right) = F_{62}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{4}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{26}\! \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_29(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_22(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_17(x))
Eq(F_14(x), F_1(x) + F_15(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_14(x)*F_4(x))
Eq(F_17(x), F_11(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_15(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_26(x) + F_7(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_7(x))
Eq(F_29(x), F_30(x) + F_43(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_14(x) + F_33(x))
Eq(F_33(x), F_30(x) + F_34(x))
Eq(F_34(x), F_35(x) + F_36(x) + F_42(x))
Eq(F_35(x), 0)
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_21(x) + F_38(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_30(x) + F_39(x))
Eq(F_42(x), F_33(x)*F_4(x))
Eq(F_43(x), F_35(x) + F_44(x) + F_55(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_25(x) + F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_50(x))
Eq(F_49(x), F_30(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_53(x) + F_54(x))
Eq(F_53(x), F_30(x))
Eq(F_54(x), F_51(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_57(x) + F_61(x))
Eq(F_57(x), F_30(x) + F_58(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_25(x))
Eq(F_61(x), F_62(x) + F_70(x))
Eq(F_62(x), F_35(x) + F_36(x) + F_63(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_65(x) + F_69(x))
Eq(F_65(x), F_30(x) + F_66(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), F_4(x)*F_68(x))
Eq(F_68(x), F_21(x))
Eq(F_69(x), F_62(x))
Eq(F_70(x), F_71(x))
Eq(F_71(x), F_4(x)*F_72(x))
Eq(F_72(x), F_26(x))
Pack JSON:
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Specification JSON:
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