0123_1302_2301_2310
Counting sequence:
1, 1, 2, 6, 20, 62, 168, 400, 855, 1675, 3058, 5270, 8658, 13664, 20840, 30864, 44557, 62901, 87058, 118390, 158480, 209154, 272504, 350912, 447075, 564031, 705186, 874342, 1075726, 1314020, 1594392, 1922528, 2304665, 2747625, 3258850, 3846438, 4519180, 5286598, 6158984, 7147440, 8263919, 9521267, 10933266, 12514678, 14281290, 16249960, 18438664, 20866544, 23553957, 26522525, 29795186, 33396246, 37351432, 41687946, 46434520, 51621472, 57280763, 63446055, 70152770, 77438150, 85341318, 93903340, 103167288, 113178304, 123983665, 135632849, 148177602, 161672006, 176172548, 191738190, 208430440, 226313424, 245453959, 265921627, 287788850, 311130966, 336026306, 362556272, 390805416, 420861520, 452815677, 486762373, 522799570, 561028790, 601555200, 644487698, 689939000, 738025728, 788868499, 842592015, 899325154, 959201062, 1022357246, 1088935668, 1159082840, 1232949920, 1310692809, 1392472249, 1478453922, 1568808550, 1663711996
Generating function in Maple syntax:
(8*x^5-20*x^4+22*x^3-16*x^2+6*x-1)/(x-1)^7
Generating function in latex syntax:
\frac{8 x^{5}-20 x^{4}+22 x^{3}-16 x^{2}+6 x -1}{\left(x -1\right)^{7}}
Generating function in sympy syntax:
(8*x**5 - 20*x**4 + 22*x**3 - 16*x**2 + 6*x - 1)/(x - 1)**7
Implicit equation for the generating function in Maple syntax:
(x-1)^7*F(x)-8*x^5+20*x^4-22*x^3+16*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right)^{7} F \! \left(x \right)-8 x^{5}+20 x^{4}-22 x^{3}+16 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
1/720*(n+1)*(n^5+20*n^4-145*n^3+520*n^2-756*n+720)
Explicit closed form in latex syntax:
\frac{\left(n +1\right) \left(n^{5}+20 n^{4}-145 n^{3}+520 n^{2}-756 n +720\right)}{720}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 62
a(n) = 1/720*(n+1)*(n^5+20*n^4-145*n^3+520*n^2-756*n+720), n >= 6
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) = 62
a \! \left(n \right) = \frac{\left(n +1\right) \left(n^{5}+20 n^{4}-145 n^{3}+520 n^{2}-756 n +720\right)}{720}, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15222/
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[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[7,x]
F[14,x] = F[15,x]+F[16,x]+F[20,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[18,x]+F[23,x]
F[18,x] = F[11,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[4,x]
F[22,x] = F[20,x]
F[23,x] = F[14,x]+F[24,x]
F[24,x] = F[25,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]+F[28,x]
F[27,x] = F[19,x]
F[28,x] = F[24,x]
F[29,x] = F[2,x]+F[30,x]
F[30,x] = F[15,x]+F[31,x]+F[35,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]+F[55,x]
F[33,x] = F[34,x]+F[7,x]
F[34,x] = F[15,x]+F[16,x]+F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,x]+F[40,x]
F[37,x] = F[38,x]+F[4,x]
F[38,x] = F[15,x]+F[20,x]+F[39,x]
F[39,x] = F[11,x]*F[4,x]
F[40,x] = F[41,x]+F[50,x]
F[41,x] = F[42,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[21,x]+F[44,x]
F[44,x] = F[41,x]+F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]+F[49,x]
F[48,x] = F[22,x]
F[49,x] = F[45,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[38,x]
F[54,x] = F[50,x]
F[55,x] = F[30,x]+F[56,x]
F[56,x] = F[15,x]+F[57,x]+F[61,x]+F[72,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[59,x]+F[73,x]
F[59,x] = F[34,x]+F[60,x]
F[60,x] = F[15,x]+F[25,x]+F[61,x]+F[72,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[63,x]+F[66,x]
F[63,x] = F[38,x]+F[64,x]
F[64,x] = F[65,x]
F[65,x] = F[19,x]*F[4,x]
F[66,x] = F[50,x]+F[67,x]
F[67,x] = F[68,x]
F[68,x] = F[4,x]*F[69,x]
F[69,x] = F[70,x]+F[71,x]
F[70,x] = F[64,x]
F[71,x] = F[67,x]
F[72,x] = 0
F[73,x] = F[56,x]+F[74,x]
F[74,x] = F[15,x]+F[75,x]+F[79,x]+F[89,x]+F[90,x]
F[75,x] = F[4,x]*F[76,x]
F[76,x] = F[77,x]+F[83,x]
F[77,x] = F[60,x]+F[78,x]
F[78,x] = F[79,x]
F[79,x] = F[4,x]*F[80,x]
F[80,x] = F[81,x]+F[82,x]
F[81,x] = F[64,x]
F[82,x] = F[67,x]
F[83,x] = F[74,x]+F[84,x]
F[84,x] = F[85,x]
F[85,x] = F[4,x]*F[86,x]
F[86,x] = F[87,x]+F[88,x]
F[87,x] = F[78,x]
F[88,x] = F[84,x]
F[89,x] = 0
F[90,x] = 0
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_{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_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{20}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{23}\! \left(x \right)
F_{18}\! \left(x \right) = F_{11}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{20}\! \left(x \right)
F_{23}\! \left(x \right) = F_{14}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \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_{28}\! \left(x \right)
F_{27}\! \left(x \right) = F_{19}\! \left(x \right)
F_{28}\! \left(x \right) = F_{24}\! \left(x \right)
F_{29}\! \left(x \right) = F_{2}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{15}\! \left(x \right)+F_{31}\! \left(x \right)+F_{35}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{55}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{7}\! \left(x \right)
F_{34}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \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_{40}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{15}\! \left(x \right)+F_{20}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{50}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{21}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{41}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{49}\! \left(x \right)
F_{48}\! \left(x \right) = F_{22}\! \left(x \right)
F_{49}\! \left(x \right) = F_{45}\! \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_{38}\! \left(x \right)
F_{54}\! \left(x \right) = F_{50}\! \left(x \right)
F_{55}\! \left(x \right) = F_{30}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{15}\! \left(x \right)+F_{57}\! \left(x \right)+F_{61}\! \left(x \right)+F_{72}\! \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_{73}\! \left(x \right)
F_{59}\! \left(x \right) = F_{34}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{15}\! \left(x \right)+F_{25}\! \left(x \right)+F_{61}\! \left(x \right)+F_{72}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)+F_{66}\! \left(x \right)
F_{63}\! \left(x \right) = F_{38}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{19}\! \left(x \right) F_{4}\! \left(x \right)
F_{66}\! \left(x \right) = F_{50}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \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_{71}\! \left(x \right)
F_{70}\! \left(x \right) = F_{64}\! \left(x \right)
F_{71}\! \left(x \right) = F_{67}\! \left(x \right)
F_{72}\! \left(x \right) = 0
F_{73}\! \left(x \right) = F_{56}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{15}\! \left(x \right)+F_{75}\! \left(x \right)+F_{79}\! \left(x \right)+F_{89}\! \left(x \right)+F_{90}\! \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_{83}\! \left(x \right)
F_{77}\! \left(x \right) = F_{60}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \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_{81}\! \left(x \right)+F_{82}\! \left(x \right)
F_{81}\! \left(x \right) = F_{64}\! \left(x \right)
F_{82}\! \left(x \right) = F_{67}\! \left(x \right)
F_{83}\! \left(x \right) = F_{74}\! \left(x \right)+F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{4}\! \left(x \right) F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{87}\! \left(x \right)+F_{88}\! \left(x \right)
F_{87}\! \left(x \right) = F_{78}\! \left(x \right)
F_{88}\! \left(x \right) = F_{84}\! \left(x \right)
F_{89}\! \left(x \right) = 0
F_{90}\! \left(x \right) = 0
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_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_7(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_20(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_23(x))
Eq(F_18(x), F_11(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_4(x))
Eq(F_22(x), F_20(x))
Eq(F_23(x), F_14(x) + F_24(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x) + F_28(x))
Eq(F_27(x), F_19(x))
Eq(F_28(x), F_24(x))
Eq(F_29(x), F_2(x) + F_30(x))
Eq(F_30(x), F_15(x) + F_31(x) + F_35(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_33(x) + F_55(x))
Eq(F_33(x), F_34(x) + F_7(x))
Eq(F_34(x), F_15(x) + F_16(x) + F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_37(x) + F_40(x))
Eq(F_37(x), F_38(x) + F_4(x))
Eq(F_38(x), F_15(x) + F_20(x) + F_39(x))
Eq(F_39(x), F_11(x)*F_4(x))
Eq(F_40(x), F_41(x) + F_50(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_21(x) + F_44(x))
Eq(F_44(x), F_41(x) + F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_49(x))
Eq(F_48(x), F_22(x))
Eq(F_49(x), F_45(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_38(x))
Eq(F_54(x), F_50(x))
Eq(F_55(x), F_30(x) + F_56(x))
Eq(F_56(x), F_15(x) + F_57(x) + F_61(x) + F_72(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_59(x) + F_73(x))
Eq(F_59(x), F_34(x) + F_60(x))
Eq(F_60(x), F_15(x) + F_25(x) + F_61(x) + F_72(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_63(x) + F_66(x))
Eq(F_63(x), F_38(x) + F_64(x))
Eq(F_64(x), F_65(x))
Eq(F_65(x), F_19(x)*F_4(x))
Eq(F_66(x), F_50(x) + F_67(x))
Eq(F_67(x), F_68(x))
Eq(F_68(x), F_4(x)*F_69(x))
Eq(F_69(x), F_70(x) + F_71(x))
Eq(F_70(x), F_64(x))
Eq(F_71(x), F_67(x))
Eq(F_72(x), 0)
Eq(F_73(x), F_56(x) + F_74(x))
Eq(F_74(x), F_15(x) + F_75(x) + F_79(x) + F_89(x) + F_90(x))
Eq(F_75(x), F_4(x)*F_76(x))
Eq(F_76(x), F_77(x) + F_83(x))
Eq(F_77(x), F_60(x) + F_78(x))
Eq(F_78(x), F_79(x))
Eq(F_79(x), F_4(x)*F_80(x))
Eq(F_80(x), F_81(x) + F_82(x))
Eq(F_81(x), F_64(x))
Eq(F_82(x), F_67(x))
Eq(F_83(x), F_74(x) + F_84(x))
Eq(F_84(x), F_85(x))
Eq(F_85(x), F_4(x)*F_86(x))
Eq(F_86(x), F_87(x) + F_88(x))
Eq(F_87(x), F_78(x))
Eq(F_88(x), F_84(x))
Eq(F_89(x), 0)
Eq(F_90(x), 0)
Pack JSON:
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Specification JSON:
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