0123_0132_2103_2310
Counting sequence:
1, 1, 2, 6, 20, 58, 128, 226, 341, 473, 622, 788, 971, 1171, 1388, 1622, 1873, 2141, 2426, 2728, 3047, 3383, 3736, 4106, 4493, 4897, 5318, 5756, 6211, 6683, 7172, 7678, 8201, 8741, 9298, 9872, 10463, 11071, 11696, 12338, 12997, 13673, 14366, 15076, 15803, 16547, 17308, 18086, 18881, 19693, 20522, 21368, 22231, 23111, 24008, 24922, 25853, 26801, 27766, 28748, 29747, 30763, 31796, 32846, 33913, 34997, 36098, 37216, 38351, 39503, 40672, 41858, 43061, 44281, 45518, 46772, 48043, 49331, 50636, 51958, 53297, 54653, 56026, 57416, 58823, 60247, 61688, 63146, 64621, 66113, 67622, 69148, 70691, 72251, 73828, 75422, 77033, 78661, 80306, 81968, 83647
Generating function in Maple syntax:
(11*x^8+4*x^7-8*x^6-14*x^5-7*x^4-2*x^3-2*x^2+2*x-1)/(x-1)^3
Generating function in latex syntax:
\frac{11 x^{8}+4 x^{7}-8 x^{6}-14 x^{5}-7 x^{4}-2 x^{3}-2 x^{2}+2 x -1}{\left(x -1\right)^{3}}
Generating function in sympy syntax:
(11*x**8 + 4*x**7 - 8*x**6 - 14*x**5 - 7*x**4 - 2*x**3 - 2*x**2 + 2*x - 1)/(x - 1)**3
Implicit equation for the generating function in Maple syntax:
-(x-1)^3*F(x)+11*x^8+4*x^7-8*x^6-14*x^5-7*x^4-2*x^3-2*x^2+2*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x -1\right)^{3} F \! \left(x \right)+11 x^{8}+4 x^{7}-8 x^{6}-14 x^{5}-7 x^{4}-2 x^{3}-2 x^{2}+2 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,n = 3,6,n = 4,20,n = 5,58,-103-25/2*n+17/2*n^2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 6 & n =3 \\ 20 & n =4 \\ 58 & n =5 \\ -103-\frac{25}{2} n +\frac{17}{2} n^{2} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 58
a(6) = 128
a(7) = 226
a(8) = 341
a(n) = -103-25/2*n+17/2*n^2, n >= 9
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(5\right) = 58
a \! \left(6\right) = 128
a \! \left(7\right) = 226
a \! \left(8\right) = 341
a \! \left(n \right) = -103-\frac{25}{2} n +\frac{17}{2} n^{2}, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14878/
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[21,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[18,x]
F[18,x] = F[19,x]*F[4,x]
F[19,x] = F[20,x]
F[20,x] = F[10,x]*F[4,x]
F[21,x] = F[2,x]+F[22,x]
F[22,x] = F[13,x]+F[23,x]+F[75,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[42,x]
F[25,x] = F[26,x]+F[7,x]
F[26,x] = F[13,x]+F[14,x]+F[27,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[29,x]+F[32,x]
F[29,x] = F[19,x]+F[30,x]
F[30,x] = F[13,x]+F[18,x]+F[31,x]
F[31,x] = F[16,x]*F[4,x]
F[32,x] = F[33,x]+F[34,x]
F[33,x] = F[13,x]+F[27,x]+F[31,x]
F[34,x] = F[13,x]+F[35,x]+F[36,x]+F[40,x]
F[35,x] = 0
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[39,x]
F[38,x] = F[18,x]
F[39,x] = F[40,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[31,x]
F[42,x] = F[43,x]+F[57,x]
F[43,x] = F[44,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]+F[52,x]
F[46,x] = F[47,x]+F[50,x]
F[47,x] = F[48,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[1,x]+F[19,x]
F[50,x] = F[51,x]
F[51,x] = F[4,x]*F[47,x]
F[52,x] = F[53,x]+F[54,x]
F[53,x] = F[44,x]
F[54,x] = F[55,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[50,x]
F[57,x] = 2*F[13,x]+F[58,x]+F[66,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[60,x]+F[61,x]
F[60,x] = F[50,x]
F[61,x] = F[62,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[64,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[19,x]+F[41,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[68,x]+F[70,x]
F[68,x] = F[63,x]+F[69,x]
F[69,x] = F[62,x]
F[70,x] = F[71,x]+F[72,x]
F[71,x] = F[66,x]
F[72,x] = F[73,x]
F[73,x] = F[4,x]*F[74,x]
F[74,x] = F[62,x]
F[75,x] = F[4,x]*F[76,x]
F[76,x] = F[77,x]+F[81,x]
F[77,x] = F[47,x]+F[78,x]
F[78,x] = F[13,x]+F[51,x]+F[79,x]
F[79,x] = F[4,x]*F[80,x]
F[80,x] = F[30,x]+F[4,x]
F[81,x] = F[82,x]+F[85,x]
F[82,x] = F[13,x]+F[75,x]+F[83,x]
F[83,x] = F[4,x]*F[84,x]
F[84,x] = F[33,x]+F[7,x]
F[85,x] = F[13,x]+F[58,x]+F[62,x]+F[86,x]
F[86,x] = F[4,x]*F[87,x]
F[87,x] = F[12,x]+F[34,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_{21}\! \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_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{2}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{13}\! \left(x \right)+F_{23}\! \left(x \right)+F_{75}\! \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_{42}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{7}\! \left(x \right)
F_{26}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \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_{29}\! \left(x \right)+F_{32}\! \left(x \right)
F_{29}\! \left(x \right) = F_{19}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{13}\! \left(x \right)+F_{18}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{34}\! \left(x \right)
F_{33}\! \left(x \right) = F_{13}\! \left(x \right)+F_{27}\! \left(x \right)+F_{31}\! \left(x \right)
F_{34}\! \left(x \right) = F_{13}\! \left(x \right)+F_{35}\! \left(x \right)+F_{36}\! \left(x \right)+F_{40}\! \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_{38}\! \left(x \right)+F_{39}\! \left(x \right)
F_{38}\! \left(x \right) = F_{18}\! \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_{31}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{57}\! \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_{52}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{50}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{54}\! \left(x \right)
F_{53}\! \left(x \right) = F_{44}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{50}\! \left(x \right)
F_{57}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{58}\! \left(x \right)+F_{66}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{61}\! \left(x \right)
F_{60}\! \left(x \right) = F_{50}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \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_{19}\! \left(x \right)+F_{41}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)+F_{70}\! \left(x \right)
F_{68}\! \left(x \right) = F_{63}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{62}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)+F_{72}\! \left(x \right)
F_{71}\! \left(x \right) = F_{66}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{4}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{62}\! \left(x \right)
F_{75}\! \left(x \right) = F_{4}\! \left(x \right) F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{77}\! \left(x \right)+F_{81}\! \left(x \right)
F_{77}\! \left(x \right) = F_{47}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{13}\! \left(x \right)+F_{51}\! \left(x \right)+F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{4}\! \left(x \right) F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{30}\! \left(x \right)+F_{4}\! \left(x \right)
F_{81}\! \left(x \right) = F_{82}\! \left(x \right)+F_{85}\! \left(x \right)
F_{82}\! \left(x \right) = F_{13}\! \left(x \right)+F_{75}\! \left(x \right)+F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{4}\! \left(x \right) F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{33}\! \left(x \right)+F_{7}\! \left(x \right)
F_{85}\! \left(x \right) = F_{13}\! \left(x \right)+F_{58}\! \left(x \right)+F_{62}\! \left(x \right)+F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{4}\! \left(x \right) F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{12}\! \left(x \right)+F_{34}\! \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_21(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_18(x))
Eq(F_18(x), F_19(x)*F_4(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_10(x)*F_4(x))
Eq(F_21(x), F_2(x) + F_22(x))
Eq(F_22(x), F_13(x) + F_23(x) + F_75(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_42(x))
Eq(F_25(x), F_26(x) + F_7(x))
Eq(F_26(x), F_13(x) + F_14(x) + F_27(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_29(x) + F_32(x))
Eq(F_29(x), F_19(x) + F_30(x))
Eq(F_30(x), F_13(x) + F_18(x) + F_31(x))
Eq(F_31(x), F_16(x)*F_4(x))
Eq(F_32(x), F_33(x) + F_34(x))
Eq(F_33(x), F_13(x) + F_27(x) + F_31(x))
Eq(F_34(x), F_13(x) + F_35(x) + F_36(x) + F_40(x))
Eq(F_35(x), 0)
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_39(x))
Eq(F_38(x), F_18(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_31(x))
Eq(F_42(x), F_43(x) + F_57(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_52(x))
Eq(F_46(x), F_47(x) + F_50(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_1(x) + F_19(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_4(x)*F_47(x))
Eq(F_52(x), F_53(x) + F_54(x))
Eq(F_53(x), F_44(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_50(x))
Eq(F_57(x), 2*F_13(x) + F_58(x) + F_66(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_60(x) + F_61(x))
Eq(F_60(x), F_50(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_19(x) + F_41(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_68(x) + F_70(x))
Eq(F_68(x), F_63(x) + F_69(x))
Eq(F_69(x), F_62(x))
Eq(F_70(x), F_71(x) + F_72(x))
Eq(F_71(x), F_66(x))
Eq(F_72(x), F_73(x))
Eq(F_73(x), F_4(x)*F_74(x))
Eq(F_74(x), F_62(x))
Eq(F_75(x), F_4(x)*F_76(x))
Eq(F_76(x), F_77(x) + F_81(x))
Eq(F_77(x), F_47(x) + F_78(x))
Eq(F_78(x), F_13(x) + F_51(x) + F_79(x))
Eq(F_79(x), F_4(x)*F_80(x))
Eq(F_80(x), F_30(x) + F_4(x))
Eq(F_81(x), F_82(x) + F_85(x))
Eq(F_82(x), F_13(x) + F_75(x) + F_83(x))
Eq(F_83(x), F_4(x)*F_84(x))
Eq(F_84(x), F_33(x) + F_7(x))
Eq(F_85(x), F_13(x) + F_58(x) + F_62(x) + F_86(x))
Eq(F_86(x), F_4(x)*F_87(x))
Eq(F_87(x), F_12(x) + F_34(x))
Pack JSON:
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Specification JSON:
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