0132_1023_1230_1302_3120
Counting sequence:
1, 1, 2, 6, 19, 50, 113, 231, 437, 776, 1307, 2105, 3263, 4894, 7133, 10139, 14097, 19220, 25751, 33965, 44171, 56714, 71977, 90383, 112397, 138528, 169331, 205409, 247415, 296054, 352085, 416323, 489641, 572972, 667311, 773717, 893315, 1027298, 1176929, 1343543, 1528549, 1733432, 1959755, 2209161, 2483375, 2784206, 3113549, 3473387, 3865793, 4292932, 4757063, 5260541, 5805819, 6395450, 7032089, 7718495, 8457533, 9252176, 10105507, 11020721, 12001127, 13050150, 14171333, 15368339, 16644953, 18005084, 19452767, 20992165, 22627571, 24363410, 26204241, 28154759, 30219797, 32404328, 34713467, 37152473, 39726751, 42441854, 45303485, 48317499, 51489905, 54826868, 58334711, 62019917, 65889131, 69949162, 74206985, 78669743, 83344749, 88239488, 93361619, 98718977, 104319575, 110171606, 116283445, 122663651, 129320969, 136264332, 143502863, 151045877, 158902883
Generating function in Maple syntax:
(x^9-3*x^8+2*x^7+3*x^6-5*x^5+8*x^4-11*x^3+11*x^2-5*x+1)/(x-1)^6
Generating function in latex syntax:
\frac{x^{9}-3 x^{8}+2 x^{7}+3 x^{6}-5 x^{5}+8 x^{4}-11 x^{3}+11 x^{2}-5 x +1}{\left(x -1\right)^{6}}
Generating function in sympy syntax:
(x**9 - 3*x**8 + 2*x**7 + 3*x**6 - 5*x**5 + 8*x**4 - 11*x**3 + 11*x**2 - 5*x + 1)/(x - 1)**6
Implicit equation for the generating function in Maple syntax:
-(x-1)^6*F(x)+x^9-3*x^8+2*x^7+3*x^6-5*x^5+8*x^4-11*x^3+11*x^2-5*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x -1\right)^{6} F \! \left(x \right)+x^{9}-3 x^{8}+2 x^{7}+3 x^{6}-5 x^{5}+8 x^{4}-11 x^{3}+11 x^{2}-5 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,n = 3,6,-7+7/12*n^3-17/12*n^2+39/10*n-1/12*n^4+1/60*n^5)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 6 & n =3 \\ -7+\frac{7}{12} n^{3}-\frac{17}{12} n^{2}+\frac{39}{10} n -\frac{1}{12} n^{4}+\frac{1}{60} n^{5} & \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) = 50
a(6) = 113
a(7) = 231
a(8) = 437
a(9) = 776
a(n) = -7+7/12*n^3-17/12*n^2+39/10*n-1/12*n^4+1/60*n^5, 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) = 50
a \! \left(6\right) = 113
a \! \left(7\right) = 231
a \! \left(8\right) = 437
a \! \left(9\right) = 776
a \! \left(n \right) = -7+\frac{7}{12} n^{3}-\frac{17}{12} n^{2}+\frac{39}{10} n -\frac{1}{12} n^{4}+\frac{1}{60} n^{5}, \quad n \geq 10
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19603/
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[34,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[26,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[14,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]+F[24,x]
F[21,x] = F[22,x]+F[4,x]
F[22,x] = F[23,x]
F[23,x] = F[21,x]*F[4,x]
F[24,x] = F[25,x]
F[25,x] = F[19,x]
F[26,x] = F[27,x]+F[28,x]+F[33,x]
F[27,x] = 0
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[21,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[27,x]+F[28,x]+F[32,x]
F[32,x] = F[14,x]*F[4,x]
F[33,x] = F[17,x]*F[4,x]
F[34,x] = F[35,x]+F[49,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[39,x]
F[38,x] = F[1,x]+F[14,x]
F[39,x] = F[35,x]+F[40,x]
F[40,x] = F[41,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[43,x]+F[48,x]
F[43,x] = F[44,x]+F[47,x]
F[44,x] = F[45,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[1,x]+F[44,x]
F[47,x] = F[19,x]
F[48,x] = F[41,x]
F[49,x] = F[27,x]+F[50,x]+F[70,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[52,x]+F[57,x]
F[52,x] = F[44,x]+F[53,x]
F[53,x] = F[27,x]+F[28,x]+F[54,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[14,x]+F[56,x]
F[56,x] = F[54,x]
F[57,x] = F[58,x]+F[62,x]
F[58,x] = F[27,x]+F[50,x]+F[59,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[35,x]+F[61,x]
F[61,x] = F[59,x]
F[62,x] = F[63,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[65,x]+F[69,x]
F[65,x] = F[66,x]
F[66,x] = F[67,x]
F[67,x] = F[4,x]*F[68,x]
F[68,x] = F[44,x]+F[66,x]
F[69,x] = F[63,x]
F[70,x] = F[4,x]*F[71,x]
F[71,x] = F[72,x]+F[78,x]
F[72,x] = F[35,x]+F[73,x]
F[73,x] = F[74,x]
F[74,x] = F[4,x]*F[75,x]
F[75,x] = F[76,x]+F[77,x]
F[76,x] = F[25,x]+F[4,x]
F[77,x] = F[74,x]
F[78,x] = F[79,x]+F[84,x]
F[79,x] = F[80,x]
F[80,x] = F[4,x]*F[81,x]
F[81,x] = F[82,x]+F[83,x]
F[82,x] = F[35,x]
F[83,x] = F[79,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[74,x]
F[88,x] = F[85,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_{34}\! \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_{26}\! \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_{14}\! \left(x \right)+F_{18}\! \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_{21}\! \left(x \right)+F_{24}\! \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_{23}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{19}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)+F_{33}\! \left(x \right)
F_{27}\! \left(x \right) = 0
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{21}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{49}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
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_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{39}\! \left(x \right) = F_{35}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{48}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{47}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{1}\! \left(x \right)+F_{44}\! \left(x \right)
F_{47}\! \left(x \right) = F_{19}\! \left(x \right)
F_{48}\! \left(x \right) = F_{41}\! \left(x \right)
F_{49}\! \left(x \right) = F_{27}\! \left(x \right)+F_{50}\! \left(x \right)+F_{70}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{57}\! \left(x \right)
F_{52}\! \left(x \right) = F_{44}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{14}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{54}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{62}\! \left(x \right)
F_{58}\! \left(x \right) = F_{27}\! \left(x \right)+F_{50}\! \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_{35}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{59}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{69}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{4}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{44}\! \left(x \right)+F_{66}\! \left(x \right)
F_{69}\! \left(x \right) = F_{63}\! \left(x \right)
F_{70}\! \left(x \right) = F_{4}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)+F_{78}\! \left(x \right)
F_{72}\! \left(x \right) = F_{35}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{4}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)+F_{77}\! \left(x \right)
F_{76}\! \left(x \right) = F_{25}\! \left(x \right)+F_{4}\! \left(x \right)
F_{77}\! \left(x \right) = F_{74}\! \left(x \right)
F_{78}\! \left(x \right) = F_{79}\! \left(x \right)+F_{84}\! \left(x \right)
F_{79}\! \left(x \right) = F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{4}\! \left(x \right) F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{82}\! \left(x \right)+F_{83}\! \left(x \right)
F_{82}\! \left(x \right) = F_{35}\! \left(x \right)
F_{83}\! \left(x \right) = F_{79}\! \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_{74}\! \left(x \right)
F_{88}\! \left(x \right) = F_{85}\! \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_34(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_26(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_14(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_21(x) + F_24(x))
Eq(F_21(x), F_22(x) + F_4(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_21(x)*F_4(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_19(x))
Eq(F_26(x), F_27(x) + F_28(x) + F_33(x))
Eq(F_27(x), 0)
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_21(x) + F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_27(x) + F_28(x) + F_32(x))
Eq(F_32(x), F_14(x)*F_4(x))
Eq(F_33(x), F_17(x)*F_4(x))
Eq(F_34(x), F_35(x) + F_49(x))
Eq(F_35(x), F_36(x))
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_1(x) + F_14(x))
Eq(F_39(x), F_35(x) + F_40(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_48(x))
Eq(F_43(x), F_44(x) + F_47(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_1(x) + F_44(x))
Eq(F_47(x), F_19(x))
Eq(F_48(x), F_41(x))
Eq(F_49(x), F_27(x) + F_50(x) + F_70(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_57(x))
Eq(F_52(x), F_44(x) + F_53(x))
Eq(F_53(x), F_27(x) + F_28(x) + F_54(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_14(x) + F_56(x))
Eq(F_56(x), F_54(x))
Eq(F_57(x), F_58(x) + F_62(x))
Eq(F_58(x), F_27(x) + F_50(x) + F_59(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_35(x) + F_61(x))
Eq(F_61(x), F_59(x))
Eq(F_62(x), F_63(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_65(x) + F_69(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), F_4(x)*F_68(x))
Eq(F_68(x), F_44(x) + F_66(x))
Eq(F_69(x), F_63(x))
Eq(F_70(x), F_4(x)*F_71(x))
Eq(F_71(x), F_72(x) + F_78(x))
Eq(F_72(x), F_35(x) + F_73(x))
Eq(F_73(x), F_74(x))
Eq(F_74(x), F_4(x)*F_75(x))
Eq(F_75(x), F_76(x) + F_77(x))
Eq(F_76(x), F_25(x) + F_4(x))
Eq(F_77(x), F_74(x))
Eq(F_78(x), F_79(x) + F_84(x))
Eq(F_79(x), F_80(x))
Eq(F_80(x), F_4(x)*F_81(x))
Eq(F_81(x), F_82(x) + F_83(x))
Eq(F_82(x), F_35(x))
Eq(F_83(x), F_79(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_74(x))
Eq(F_88(x), F_85(x))
Pack JSON:
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Specification JSON:
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