0132_0213_1203_2301_3201
Counting sequence:
1, 1, 2, 6, 19, 51, 116, 232, 421, 709, 1126, 1706, 2487, 3511, 4824, 6476, 8521, 11017, 14026, 17614, 21851, 26811, 32572, 39216, 46829, 55501, 65326, 76402, 88831, 102719, 118176, 135316, 154257, 175121, 198034, 223126, 250531, 280387, 312836, 348024, 386101, 427221, 471542, 519226, 570439, 625351, 684136, 746972, 814041, 885529, 961626, 1042526, 1128427, 1219531, 1316044, 1418176, 1526141, 1640157, 1760446, 1887234, 2020751, 2161231, 2308912, 2464036, 2626849, 2797601, 2976546, 3163942, 3360051, 3565139, 3779476, 4003336, 4236997, 4480741, 4734854, 4999626, 5275351, 5562327, 5860856, 6171244, 6493801, 6828841, 7176682, 7537646, 7912059, 8300251, 8702556, 9119312, 9550861, 9997549, 10459726, 10937746, 11431967, 11942751, 12470464, 13015476, 13578161, 14158897, 14758066, 15376054, 16013251
Generating function in Maple syntax:
-(4*x^4-4*x^3+7*x^2-4*x+1)/(-1+x)^5
Generating function in latex syntax:
-\frac{4 x^{4}-4 x^{3}+7 x^{2}-4 x +1}{\left(-1+x \right)^{5}}
Generating function in sympy syntax:
(-4*x**4 + 4*x**3 - 7*x**2 + 4*x - 1)/(x - 1)**5
Implicit equation for the generating function in Maple syntax:
(-1+x)^5*F(x)+4*x^4-4*x^3+7*x^2-4*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(-1+x \right)^{5} F \! \left(x \right)+4 x^{4}-4 x^{3}+7 x^{2}-4 x +1 = 0
Explicit closed form in Maple syntax:
1-2/3*n^3+4/3*n^2-5/6*n+1/6*n^4
Explicit closed form in latex syntax:
1-\frac{2}{3} n^{3}+\frac{4}{3} n^{2}-\frac{5}{6} n +\frac{1}{6} n^{4}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(n) = 1-2/3*n^3+4/3*n^2-5/6*n+1/6*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) = 19
a \! \left(n \right) = 1-\frac{2}{3} n^{3}+\frac{4}{3} n^{2}-\frac{5}{6} n +\frac{1}{6} n^{4}, \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18433/
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[24,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[23,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[10,x]+F[17,x]
F[17,x] = F[18,x]+F[21,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[1,x]+F[18,x]
F[21,x] = F[22,x]
F[22,x] = F[17,x]*F[4,x]
F[23,x] = F[22,x]
F[24,x] = F[25,x]+F[42,x]
F[25,x] = F[26,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[28,x]+F[29,x]
F[28,x] = F[1,x]+F[14,x]
F[29,x] = F[18,x]+F[30,x]
F[30,x] = F[31,x]+F[32,x]+F[34,x]
F[31,x] = 0
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[14,x]+F[30,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[17,x]+F[36,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[18,x]+F[37,x]
F[40,x] = F[41,x]
F[41,x] = F[36,x]*F[4,x]
F[42,x] = F[31,x]+F[43,x]+F[66,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[45,x]+F[62,x]
F[45,x] = F[46,x]+F[7,x]
F[46,x] = F[31,x]+F[47,x]+F[55,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]+F[52,x]
F[49,x] = F[11,x]+F[50,x]
F[50,x] = F[51,x]
F[51,x] = F[4,x]*F[49,x]
F[52,x] = F[21,x]+F[53,x]
F[53,x] = F[54,x]
F[54,x] = F[4,x]*F[52,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[17,x]+F[57,x]
F[57,x] = F[58,x]+F[61,x]
F[58,x] = F[31,x]+F[34,x]+F[59,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[11,x]+F[21,x]
F[61,x] = F[41,x]
F[62,x] = F[63,x]+F[70,x]
F[63,x] = F[31,x]+F[64,x]+F[66,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[63,x]+F[7,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[17,x]+F[68,x]
F[68,x] = F[30,x]+F[69,x]
F[69,x] = F[41,x]
F[70,x] = F[31,x]+F[71,x]+F[73,x]+F[78,x]
F[71,x] = F[4,x]*F[72,x]
F[72,x] = F[46,x]+F[70,x]
F[73,x] = F[4,x]*F[74,x]
F[74,x] = F[52,x]+F[75,x]
F[75,x] = F[40,x]+F[76,x]
F[76,x] = F[77,x]
F[77,x] = F[4,x]*F[75,x]
F[78,x] = F[4,x]*F[79,x]
F[79,x] = F[36,x]+F[80,x]
F[80,x] = F[81,x]+F[92,x]
F[81,x] = 2*F[31,x]+F[82,x]+F[84,x]
F[82,x] = F[4,x]*F[83,x]
F[83,x] = F[21,x]+F[40,x]
F[84,x] = F[4,x]*F[85,x]
F[85,x] = F[36,x]+F[86,x]
F[86,x] = F[87,x]+F[90,x]
F[87,x] = F[88,x]
F[88,x] = F[4,x]*F[89,x]
F[89,x] = F[37,x]+F[87,x]
F[90,x] = F[91,x]
F[91,x] = F[4,x]*F[86,x]
F[92,x] = F[91,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_{24}\! \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_{23}\! \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_{10}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{21}\! \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_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{22}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{42}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{29}\! \left(x \right)
F_{28}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{29}\! \left(x \right) = F_{18}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)+F_{34}\! \left(x \right)
F_{31}\! \left(x \right) = 0
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{14}\! \left(x \right)+F_{30}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{17}\! \left(x \right)+F_{36}\! \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_{18}\! \left(x \right)+F_{37}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{42}\! \left(x \right) = F_{31}\! \left(x \right)+F_{43}\! \left(x \right)+F_{66}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{62}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{7}\! \left(x \right)
F_{46}\! \left(x \right) = F_{31}\! \left(x \right)+F_{47}\! \left(x \right)+F_{55}\! \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_{52}\! \left(x \right)
F_{49}\! \left(x \right) = F_{11}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{52}\! \left(x \right) = F_{21}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{17}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{61}\! \left(x \right)
F_{58}\! \left(x \right) = F_{31}\! \left(x \right)+F_{34}\! \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_{11}\! \left(x \right)+F_{21}\! \left(x \right)
F_{61}\! \left(x \right) = F_{41}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)+F_{70}\! \left(x \right)
F_{63}\! \left(x \right) = F_{31}\! \left(x \right)+F_{64}\! \left(x \right)+F_{66}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{63}\! \left(x \right)+F_{7}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{17}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{30}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{41}\! \left(x \right)
F_{70}\! \left(x \right) = F_{31}\! \left(x \right)+F_{71}\! \left(x \right)+F_{73}\! \left(x \right)+F_{78}\! \left(x \right)
F_{71}\! \left(x \right) = F_{4}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{46}\! \left(x \right)+F_{70}\! \left(x \right)
F_{73}\! \left(x \right) = F_{4}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{52}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{40}\! \left(x \right)+F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{4}\! \left(x \right) F_{75}\! \left(x \right)
F_{78}\! \left(x \right) = F_{4}\! \left(x \right) F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{36}\! \left(x \right)+F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{81}\! \left(x \right)+F_{92}\! \left(x \right)
F_{81}\! \left(x \right) = 2 F_{31}\! \left(x \right)+F_{82}\! \left(x \right)+F_{84}\! \left(x \right)
F_{82}\! \left(x \right) = F_{4}\! \left(x \right) F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{21}\! \left(x \right)+F_{40}\! \left(x \right)
F_{84}\! \left(x \right) = F_{4}\! \left(x \right) F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{36}\! \left(x \right)+F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{87}\! \left(x \right)+F_{90}\! \left(x \right)
F_{87}\! \left(x \right) = F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{4}\! \left(x \right) F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = F_{37}\! \left(x \right)+F_{87}\! \left(x \right)
F_{90}\! \left(x \right) = F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{4}\! \left(x \right) F_{86}\! \left(x \right)
F_{92}\! \left(x \right) = F_{91}\! \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_24(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_23(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_10(x) + F_17(x))
Eq(F_17(x), F_18(x) + F_21(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_1(x) + F_18(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_17(x)*F_4(x))
Eq(F_23(x), F_22(x))
Eq(F_24(x), F_25(x) + F_42(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_28(x) + F_29(x))
Eq(F_28(x), F_1(x) + F_14(x))
Eq(F_29(x), F_18(x) + F_30(x))
Eq(F_30(x), F_31(x) + F_32(x) + F_34(x))
Eq(F_31(x), 0)
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_14(x) + F_30(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_17(x) + F_36(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_18(x) + F_37(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_36(x)*F_4(x))
Eq(F_42(x), F_31(x) + F_43(x) + F_66(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_62(x))
Eq(F_45(x), F_46(x) + F_7(x))
Eq(F_46(x), F_31(x) + F_47(x) + F_55(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_52(x))
Eq(F_49(x), F_11(x) + F_50(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_4(x)*F_49(x))
Eq(F_52(x), F_21(x) + F_53(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_4(x)*F_52(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_17(x) + F_57(x))
Eq(F_57(x), F_58(x) + F_61(x))
Eq(F_58(x), F_31(x) + F_34(x) + F_59(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_11(x) + F_21(x))
Eq(F_61(x), F_41(x))
Eq(F_62(x), F_63(x) + F_70(x))
Eq(F_63(x), F_31(x) + F_64(x) + F_66(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_63(x) + F_7(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_17(x) + F_68(x))
Eq(F_68(x), F_30(x) + F_69(x))
Eq(F_69(x), F_41(x))
Eq(F_70(x), F_31(x) + F_71(x) + F_73(x) + F_78(x))
Eq(F_71(x), F_4(x)*F_72(x))
Eq(F_72(x), F_46(x) + F_70(x))
Eq(F_73(x), F_4(x)*F_74(x))
Eq(F_74(x), F_52(x) + F_75(x))
Eq(F_75(x), F_40(x) + F_76(x))
Eq(F_76(x), F_77(x))
Eq(F_77(x), F_4(x)*F_75(x))
Eq(F_78(x), F_4(x)*F_79(x))
Eq(F_79(x), F_36(x) + F_80(x))
Eq(F_80(x), F_81(x) + F_92(x))
Eq(F_81(x), 2*F_31(x) + F_82(x) + F_84(x))
Eq(F_82(x), F_4(x)*F_83(x))
Eq(F_83(x), F_21(x) + F_40(x))
Eq(F_84(x), F_4(x)*F_85(x))
Eq(F_85(x), F_36(x) + F_86(x))
Eq(F_86(x), F_87(x) + F_90(x))
Eq(F_87(x), F_88(x))
Eq(F_88(x), F_4(x)*F_89(x))
Eq(F_89(x), F_37(x) + F_87(x))
Eq(F_90(x), F_91(x))
Eq(F_91(x), F_4(x)*F_86(x))
Eq(F_92(x), F_91(x))
Pack JSON:
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Specification JSON:
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