0123_0132_1230_1320_3120
Counting sequence:
1, 1, 2, 6, 19, 55, 135, 287, 546, 954, 1560, 2420, 3597, 5161, 7189, 9765, 12980, 16932, 21726, 27474, 34295, 42315, 51667, 62491, 74934, 89150, 105300, 123552, 144081, 167069, 192705, 221185, 252712, 287496, 325754, 367710, 413595, 463647, 518111, 577239, 641290, 710530, 785232, 865676, 952149, 1044945, 1144365, 1250717, 1364316, 1485484, 1614550, 1751850, 1897727, 2052531, 2216619, 2390355, 2574110, 2768262, 2973196, 3189304, 3416985, 3656645, 3908697, 4173561, 4451664, 4743440, 5049330, 5369782, 5705251, 6056199, 6423095, 6806415, 7206642, 7624266, 8059784, 8513700, 8986525, 9478777, 9990981, 10523669, 11077380, 11652660, 12250062, 12870146, 13513479, 14180635, 14872195, 15588747, 16330886, 17099214, 17894340, 18716880, 19567457, 20446701, 21355249, 22293745, 23262840, 24263192, 25295466, 26360334, 27458475
Generating function in Maple syntax:
(x^6-4*x^5-4*x^4+4*x^3-7*x^2+4*x-1)/(-1+x)^5
Generating function in latex syntax:
\frac{x^{6}-4 x^{5}-4 x^{4}+4 x^{3}-7 x^{2}+4 x -1}{\left(-1+x \right)^{5}}
Generating function in sympy syntax:
(x**6 - 4*x**5 - 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)+x^6-4*x^5-4*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)+x^{6}-4 x^{5}-4 x^{4}+4 x^{3}-7 x^{2}+4 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/24*n*(-1+n)*(7*n^2-35*n+66))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{n \left(-1+n \right) \left(7 n^{2}-35 n +66\right)}{24} & \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) = 55
a(6) = 135
a(n) = 1/24*n*(-1+n)*(7*n^2-35*n+66), n >= 7
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) = 55
a \! \left(6\right) = 135
a \! \left(n \right) = \frac{n \left(-1+n \right) \left(7 n^{2}-35 n +66\right)}{24}, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16514/
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[27,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[21,x]
F[12,x] = F[13,x]
F[13,x] = F[14,x]*F[4,x]
F[14,x] = F[10,x]+F[15,x]
F[15,x] = F[12,x]+F[16,x]
F[16,x] = F[17,x]
F[17,x] = F[18,x]*F[4,x]
F[18,x] = F[19,x]+F[20,x]
F[19,x] = F[4,x]
F[20,x] = F[16,x]
F[21,x] = F[22,x]+F[23,x]+F[26,x]
F[22,x] = 0
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[19,x]+F[25,x]
F[25,x] = F[21,x]
F[26,x] = F[12,x]*F[4,x]
F[27,x] = F[28,x]+F[44,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[31,x]+F[32,x]
F[31,x] = F[1,x]+F[12,x]
F[32,x] = F[28,x]+F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[36,x]+F[38,x]
F[36,x] = F[12,x]+F[37,x]
F[37,x] = F[17,x]
F[38,x] = F[33,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[37,x]
F[43,x] = F[39,x]
F[44,x] = F[22,x]+F[45,x]+F[77,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]+F[52,x]
F[47,x] = F[48,x]+F[7,x]
F[48,x] = F[22,x]+F[23,x]+F[49,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[15,x]+F[51,x]
F[51,x] = F[37,x]
F[52,x] = F[44,x]+F[53,x]
F[53,x] = 2*F[22,x]+F[54,x]+F[68,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[56,x]+F[60,x]
F[56,x] = F[48,x]+F[57,x]
F[57,x] = F[58,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[37,x]
F[60,x] = F[53,x]+F[61,x]
F[61,x] = 3*F[22,x]+F[62,x]+F[66,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[64,x]+F[65,x]
F[64,x] = F[57,x]
F[65,x] = F[61,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[39,x]
F[68,x] = F[4,x]*F[69,x]
F[69,x] = F[70,x]+F[76,x]
F[70,x] = F[33,x]+F[71,x]
F[71,x] = F[72,x]
F[72,x] = F[4,x]*F[73,x]
F[73,x] = F[74,x]+F[75,x]
F[74,x] = F[16,x]
F[75,x] = F[71,x]
F[76,x] = F[39,x]
F[77,x] = F[4,x]*F[78,x]
F[78,x] = F[79,x]+F[85,x]
F[79,x] = F[28,x]+F[80,x]
F[80,x] = F[81,x]
F[81,x] = F[4,x]*F[82,x]
F[82,x] = F[83,x]+F[84,x]
F[83,x] = F[16,x]+F[4,x]
F[84,x] = F[71,x]+F[80,x]
F[85,x] = F[33,x]+F[86,x]
F[86,x] = F[87,x]
F[87,x] = F[4,x]*F[88,x]
F[88,x] = F[89,x]+F[90,x]
F[89,x] = F[21,x]
F[90,x] = F[86,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_{27}\! \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_{21}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{10}\! \left(x \right)+F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right)+F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)
F_{19}\! \left(x \right) = F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{16}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{26}\! \left(x \right)
F_{22}\! \left(x \right) = 0
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{19}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{21}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{44}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)
F_{31}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{32}\! \left(x \right) = F_{28}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{38}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{17}\! \left(x \right)
F_{38}\! \left(x \right) = F_{33}\! \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_{37}\! \left(x \right)
F_{43}\! \left(x \right) = F_{39}\! \left(x \right)
F_{44}\! \left(x \right) = F_{22}\! \left(x \right)+F_{45}\! \left(x \right)+F_{77}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{52}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{7}\! \left(x \right)
F_{48}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \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_{15}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{37}\! \left(x \right)
F_{52}\! \left(x \right) = F_{44}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{54}\! \left(x \right)+F_{68}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{60}\! \left(x \right)
F_{56}\! \left(x \right) = F_{48}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{37}\! \left(x \right)
F_{60}\! \left(x \right) = F_{53}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = 3 F_{22}\! \left(x \right)+F_{62}\! \left(x \right)+F_{66}\! \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_{65}\! \left(x \right)
F_{64}\! \left(x \right) = F_{57}\! \left(x \right)
F_{65}\! \left(x \right) = F_{61}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{39}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)+F_{76}\! \left(x \right)
F_{70}\! \left(x \right) = F_{33}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{4}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)+F_{75}\! \left(x \right)
F_{74}\! \left(x \right) = F_{16}\! \left(x \right)
F_{75}\! \left(x \right) = F_{71}\! \left(x \right)
F_{76}\! \left(x \right) = F_{39}\! \left(x \right)
F_{77}\! \left(x \right) = F_{4}\! \left(x \right) F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{79}\! \left(x \right)+F_{85}\! \left(x \right)
F_{79}\! \left(x \right) = F_{28}\! \left(x \right)+F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{4}\! \left(x \right) F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{83}\! \left(x \right)+F_{84}\! \left(x \right)
F_{83}\! \left(x \right) = F_{16}\! \left(x \right)+F_{4}\! \left(x \right)
F_{84}\! \left(x \right) = F_{71}\! \left(x \right)+F_{80}\! \left(x \right)
F_{85}\! \left(x \right) = F_{33}\! \left(x \right)+F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{4}\! \left(x \right) F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{89}\! \left(x \right)+F_{90}\! \left(x \right)
F_{89}\! \left(x \right) = F_{21}\! \left(x \right)
F_{90}\! \left(x \right) = F_{86}\! \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_27(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_21(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_14(x)*F_4(x))
Eq(F_14(x), F_10(x) + F_15(x))
Eq(F_15(x), F_12(x) + F_16(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_18(x)*F_4(x))
Eq(F_18(x), F_19(x) + F_20(x))
Eq(F_19(x), F_4(x))
Eq(F_20(x), F_16(x))
Eq(F_21(x), F_22(x) + F_23(x) + F_26(x))
Eq(F_22(x), 0)
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_19(x) + F_25(x))
Eq(F_25(x), F_21(x))
Eq(F_26(x), F_12(x)*F_4(x))
Eq(F_27(x), F_28(x) + F_44(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_31(x) + F_32(x))
Eq(F_31(x), F_1(x) + F_12(x))
Eq(F_32(x), F_28(x) + F_33(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_36(x) + F_38(x))
Eq(F_36(x), F_12(x) + F_37(x))
Eq(F_37(x), F_17(x))
Eq(F_38(x), F_33(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_37(x))
Eq(F_43(x), F_39(x))
Eq(F_44(x), F_22(x) + F_45(x) + F_77(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_52(x))
Eq(F_47(x), F_48(x) + F_7(x))
Eq(F_48(x), F_22(x) + F_23(x) + F_49(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_15(x) + F_51(x))
Eq(F_51(x), F_37(x))
Eq(F_52(x), F_44(x) + F_53(x))
Eq(F_53(x), 2*F_22(x) + F_54(x) + F_68(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_60(x))
Eq(F_56(x), F_48(x) + F_57(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_37(x))
Eq(F_60(x), F_53(x) + F_61(x))
Eq(F_61(x), 3*F_22(x) + F_62(x) + F_66(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_64(x) + F_65(x))
Eq(F_64(x), F_57(x))
Eq(F_65(x), F_61(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_39(x))
Eq(F_68(x), F_4(x)*F_69(x))
Eq(F_69(x), F_70(x) + F_76(x))
Eq(F_70(x), F_33(x) + F_71(x))
Eq(F_71(x), F_72(x))
Eq(F_72(x), F_4(x)*F_73(x))
Eq(F_73(x), F_74(x) + F_75(x))
Eq(F_74(x), F_16(x))
Eq(F_75(x), F_71(x))
Eq(F_76(x), F_39(x))
Eq(F_77(x), F_4(x)*F_78(x))
Eq(F_78(x), F_79(x) + F_85(x))
Eq(F_79(x), F_28(x) + F_80(x))
Eq(F_80(x), F_81(x))
Eq(F_81(x), F_4(x)*F_82(x))
Eq(F_82(x), F_83(x) + F_84(x))
Eq(F_83(x), F_16(x) + F_4(x))
Eq(F_84(x), F_71(x) + F_80(x))
Eq(F_85(x), F_33(x) + F_86(x))
Eq(F_86(x), F_87(x))
Eq(F_87(x), F_4(x)*F_88(x))
Eq(F_88(x), F_89(x) + F_90(x))
Eq(F_89(x), F_21(x))
Eq(F_90(x), F_86(x))
Pack JSON:
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Specification JSON:
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