0123_0132_1203_1320_2130
Counting sequence:
1, 1, 2, 6, 19, 54, 138, 323, 712, 1505, 3086, 6189, 12208, 23778, 45861, 87770, 166936, 315908, 595343, 1118085, 2093755, 3911253, 7291293, 13568237, 25210537, 46781324, 86710622, 160563720, 297066823, 549212892, 1014725586, 1873758377, 3458331208, 6380201651, 11766311042, 21692249079, 39980185564, 73667573376, 135710258133, 249957093416, 460304249824, 847540000958, 1560339066923, 2872289438151, 5286812347231, 9730190811867, 17906686395813, 32951833310723, 60634248070345, 111566449082078, 205271749317602, 377665346626866, 694815664034899, 1278257779141482, 2351555927970498, 4325951528472887, 7957904531074624, 14638873440330125, 26928330248175710, 49534170420686697, 91116037058592736, 167602262877957318, 308290858457134701, 567071271644080286, 1043064894329461024, 1918589639031355328, 3528988920955859567, 6491069184868312881, 11939336591358122203, 21960509274236520393, 40392718474019769813, 74295482340225446465, 136653431512649578849, 251349271751673664952, 462310546453495395662, 850333249991544207876, 1564025429319385535287, 2876721587160822966816, 5291164988990822797338, 9732049089386499207149, 17900157471972682958512, 32923730440700010849839, 60556517698843736881250, 111381345198650980433667, 204863113622113821765820, 376803436390662182416884, 693051875366399721544365, 1274724865405202105981756, 2344590597343263127115992, 4312384198321890452059850, 7931726941458380299738139, 14588745877718583990901827, 26832928438481929469257519, 49353516819237018598432095, 90775378117998731623672557, 166962125919857004094966439, 307091510383793278285744753, 564829806492488862376649690, 1038884724393679517098288226, 1910808114938341878473863422, 3514526001090430851002896627
Generating function in Maple syntax:
-(x^9-x^8-3*x^7-x^6-2*x^5+x^3+5*x^2-4*x+1)/(x^2+x-1)/(x^3+x^2+x-1)/(-1+x)^3
Generating function in latex syntax:
-\frac{x^{9}-x^{8}-3 x^{7}-x^{6}-2 x^{5}+x^{3}+5 x^{2}-4 x +1}{\left(x^{2}+x -1\right) \left(x^{3}+x^{2}+x -1\right) \left(-1+x \right)^{3}}
Generating function in sympy syntax:
(-x**9 + x**8 + 3*x**7 + x**6 + 2*x**5 - x**3 - 5*x**2 + 4*x - 1)/((x - 1)**3*(x**2 + x - 1)*(x**3 + x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(x^2+x-1)*(x^3+x^2+x-1)*(-1+x)^3*F(x)+x^9-x^8-3*x^7-x^6-2*x^5+x^3+5*x^2-4*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}+x -1\right) \left(x^{3}+x^{2}+x -1\right) \left(-1+x \right)^{3} F \! \left(x \right)+x^{9}-x^{8}-3 x^{7}-x^{6}-2 x^{5}+x^{3}+5 x^{2}-4 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,0,1/352*(((-116*11^(1/2)+440*I)*3^(1/2)-348*I*11^(1/2)+440)*(17+3*11^(1/2)*3^(1/2))^(1/3)+1408+((163*11^(1/2)+1001*I)*3^(1/2)-489*I*11^(1/2)-1001)*(17+3*11^(1/2)*3^(1/2))^(2/3))*(1/24*((17*I+3*11^(1/2))*3^(1/2)-9*I*11^(1/2)-17)*(17+3*11^(1/2)*3^(1/2))^(2/3)-1/6*I*3^(1/2)*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/6*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/3)^(-n)+1/352*(((163*11^(1/2)-1001*I)*3^(1/2)+489*I*11^(1/2)-1001)*(17+3*11^(1/2)*3^(1/2))^(2/3)+1408+((-116*11^(1/2)-440*I)*3^(1/2)+348*I*11^(1/2)+440)*(17+3*11^(1/2)*3^(1/2))^(1/3))*(1/24*((-17*I+3*11^(1/2))*3^(1/2)+9*I*11^(1/2)-17)*(17+3*11^(1/2)*3^(1/2))^(2/3)+1/6*I*3^(1/2)*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/6*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/3)^(-n)+1/352*((-326*11^(1/2)*3^(1/2)+2002)*(17+3*11^(1/2)*3^(1/2))^(2/3)+232*(17+3*11^(1/2)*3^(1/2))^(1/3)*11^(1/2)*3^(1/2)-880*(17+3*11^(1/2)*3^(1/2))^(1/3)+1408)*(1/3*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/3+17/12*(17+3*11^(1/2)*3^(1/2))^(2/3)-1/4*(17+3*11^(1/2)*3^(1/2))^(2/3)*11^(1/2)*3^(1/2))^(-n)+1/352*(1408*5^(1/2)-3168)*(-1/2*5^(1/2)-1/2)^(-n)+1/352*(-1408*5^(1/2)-3168)*(1/2*5^(1/2)-1/2)^(-n)-3/4*n^2+9/4*n+7)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}0 & n =0 \\ \frac{\left(\left(\left(-116 \sqrt{11}+440 \,\mathrm{I}\right) \sqrt{3}-348 \,\mathrm{I} \sqrt{11}+440\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+1408+\left(\left(163 \sqrt{11}+1001 \,\mathrm{I}\right) \sqrt{3}-489 \,\mathrm{I} \sqrt{11}-1001\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}-9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}-\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{352}\\+\\\frac{\left(\left(\left(163 \sqrt{11}-1001 \,\mathrm{I}\right) \sqrt{3}+489 \,\mathrm{I} \sqrt{11}-1001\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+1408+\left(\left(-116 \sqrt{11}-440 \,\mathrm{I}\right) \sqrt{3}+348 \,\mathrm{I} \sqrt{11}+440\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{\left(\left(-17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}+9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}+\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{352}\\+\\\frac{\left(\left(-326 \sqrt{11}\, \sqrt{3}+2002\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+232 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{11}\, \sqrt{3}-880 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+1408\right) \left(\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}-\frac{1}{3}+\frac{17 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{12}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{11}\, \sqrt{3}}{4}\right)^{-n}}{352}\\+\frac{\left(1408 \sqrt{5}-3168\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{352}+\frac{\left(-1408 \sqrt{5}-3168\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{352}\\-\frac{3 n^{2}}{4}+\frac{9 n}{4}+7 & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 54
a(6) = 138
a(7) = 323
a(8) = 712
a(9) = 1505
a(n+5) = -3/2*n^2-a(n)-2*a(n+1)-a(n+2)+a(n+3)+2*a(n+4)+15/2*n+17, n >= 10
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) = 54
a \! \left(6\right) = 138
a \! \left(7\right) = 323
a \! \left(8\right) = 712
a \! \left(9\right) = 1505
a \! \left(n +5\right) = -\frac{3 n^{2}}{2}-a \! \left(n \right)-2 a \! \left(n +1\right)-a \! \left(n +2\right)+a \! \left(n +3\right)+2 a \! \left(n +4\right)+\frac{15 n}{2}+17, \quad n \geq 10
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16426/
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[19,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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[7,x]
F[12,x] = F[13,x]+F[14,x]+F[18,x]
F[13,x] = 0
F[14,x] = F[15,x]*F[4,x]
F[15,x] = F[16,x]+F[17,x]
F[16,x] = F[4,x]
F[17,x] = F[14,x]
F[18,x] = F[4,x]*F[7,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[13,x]+F[21,x]+F[57,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[33,x]
F[23,x] = F[24,x]+F[7,x]
F[24,x] = F[13,x]+F[14,x]+F[25,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]+F[29,x]
F[27,x] = F[28,x]+F[7,x]
F[28,x] = F[18,x]
F[29,x] = F[30,x]+F[31,x]
F[30,x] = F[14,x]
F[31,x] = F[32,x]
F[32,x] = F[30,x]*F[4,x]
F[33,x] = F[34,x]+F[48,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,x]+F[38,x]
F[37,x] = F[30,x]+F[7,x]
F[38,x] = F[34,x]+F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]+F[43,x]
F[42,x] = F[30,x]
F[43,x] = F[44,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]+F[47,x]
F[46,x] = F[30,x]
F[47,x] = F[44,x]
F[48,x] = F[49,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[51,x]+F[52,x]
F[51,x] = F[24,x]
F[52,x] = F[53,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[55,x]+F[56,x]
F[55,x] = F[24,x]
F[56,x] = F[53,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[59,x]+F[62,x]
F[59,x] = F[2,x]+F[60,x]
F[60,x] = F[61,x]
F[61,x] = F[2,x]*F[4,x]
F[62,x] = F[34,x]+F[63,x]
F[63,x] = F[64,x]
F[64,x] = F[34,x]*F[4,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_{19}\! \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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{7}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{13}\! \left(x \right) = 0
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)
F_{16}\! \left(x \right) = F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right)
F_{18}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{13}\! \left(x \right)+F_{21}\! \left(x \right)+F_{57}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{33}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{7}\! \left(x \right)
F_{24}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{29}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{7}\! \left(x \right)
F_{28}\! \left(x \right) = F_{18}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{31}\! \left(x \right)
F_{30}\! \left(x \right) = F_{14}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{48}\! \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_{37}\! \left(x \right)+F_{38}\! \left(x \right)
F_{37}\! \left(x \right) = F_{30}\! \left(x \right)+F_{7}\! \left(x \right)
F_{38}\! \left(x \right) = F_{34}\! \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_{42}\! \left(x \right)+F_{43}\! \left(x \right)
F_{42}\! \left(x \right) = F_{30}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{47}\! \left(x \right)
F_{46}\! \left(x \right) = F_{30}\! \left(x \right)
F_{47}\! \left(x \right) = F_{44}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{52}\! \left(x \right)
F_{51}\! \left(x \right) = F_{24}\! \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_{55}\! \left(x \right)+F_{56}\! \left(x \right)
F_{55}\! \left(x \right) = F_{24}\! \left(x \right)
F_{56}\! \left(x \right) = F_{53}\! \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_{2}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{62}\! \left(x \right) = F_{34}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \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_19(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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_7(x))
Eq(F_12(x), F_13(x) + F_14(x) + F_18(x))
Eq(F_13(x), 0)
Eq(F_14(x), F_15(x)*F_4(x))
Eq(F_15(x), F_16(x) + F_17(x))
Eq(F_16(x), F_4(x))
Eq(F_17(x), F_14(x))
Eq(F_18(x), F_4(x)*F_7(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_13(x) + F_21(x) + F_57(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_33(x))
Eq(F_23(x), F_24(x) + F_7(x))
Eq(F_24(x), F_13(x) + F_14(x) + F_25(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x) + F_29(x))
Eq(F_27(x), F_28(x) + F_7(x))
Eq(F_28(x), F_18(x))
Eq(F_29(x), F_30(x) + F_31(x))
Eq(F_30(x), F_14(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_30(x)*F_4(x))
Eq(F_33(x), F_34(x) + F_48(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_37(x) + F_38(x))
Eq(F_37(x), F_30(x) + F_7(x))
Eq(F_38(x), F_34(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_42(x) + F_43(x))
Eq(F_42(x), F_30(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_47(x))
Eq(F_46(x), F_30(x))
Eq(F_47(x), F_44(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_52(x))
Eq(F_51(x), F_24(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_56(x))
Eq(F_55(x), F_24(x))
Eq(F_56(x), F_53(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_2(x) + F_60(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_2(x)*F_4(x))
Eq(F_62(x), F_34(x) + F_63(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_34(x)*F_4(x))
Pack JSON:
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Specification JSON:
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