0123_1302_1320_2310
Counting sequence:
1, 1, 2, 6, 20, 61, 155, 334, 637, 1110, 1806, 2785, 4114, 5867, 8125, 10976, 14515, 18844, 24072, 30315, 37696, 46345, 56399, 68002, 81305, 96466, 113650, 133029, 154782, 179095, 206161, 236180, 269359, 305912, 346060, 390031, 438060, 490389, 547267, 608950, 675701, 747790, 825494, 909097, 998890, 1095171, 1198245, 1308424, 1426027, 1551380, 1684816, 1826675, 1977304, 2137057, 2306295, 2485386, 2674705, 2874634, 3085562, 3307885, 3542006, 3788335, 4047289, 4319292, 4604775, 4904176, 5217940, 5546519, 5890372, 6249965, 6625771, 7018270, 7427949, 7855302, 8300830, 8765041, 9248450, 9751579, 10274957, 10819120, 11384611, 11971980, 12581784, 13214587, 13870960, 14551481, 15256735, 15987314, 16743817, 17526850, 18337026, 19174965, 20041294, 20936647, 21861665, 22816996, 23803295, 24821224, 25871452, 26954655, 28071516
Generating function in Maple syntax:
-(x^8-3*x^7-x^6+5*x^5+5*x^4-4*x^3+7*x^2-4*x+1)/(x-1)^5
Generating function in latex syntax:
-\frac{x^{8}-3 x^{7}-x^{6}+5 x^{5}+5 x^{4}-4 x^{3}+7 x^{2}-4 x +1}{\left(x -1\right)^{5}}
Generating function in sympy syntax:
(-x**8 + 3*x**7 + x**6 - 5*x**5 - 5*x**4 + 4*x**3 - 7*x**2 + 4*x - 1)/(x - 1)**5
Implicit equation for the generating function in Maple syntax:
(x-1)^5*F(x)+x^8-3*x^7-x^6+5*x^5+5*x^4-4*x^3+7*x^2-4*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right)^{5} F \! \left(x \right)+x^{8}-3 x^{7}-x^{6}+5 x^{5}+5 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,n = 2,2,n = 3,6,-9-13/12*n^3-31/24*n^2+133/12*n+7/24*n^4)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 6 & n =3 \\ -9-\frac{13}{12} n^{3}-\frac{31}{24} n^{2}+\frac{133}{12} n +\frac{7}{24} n^{4} & \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) = 61
a(6) = 155
a(7) = 334
a(8) = 637
a(n) = 1/24*(-1+n)*(7*n^3-19*n^2-50*n+216), 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) = 61
a \! \left(6\right) = 155
a \! \left(7\right) = 334
a \! \left(8\right) = 637
a \! \left(n \right) = \frac{\left(-1+n \right) \left(7 n^{3}-19 n^{2}-50 n +216\right)}{24}, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15208/
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[33,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[28,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[18,x]+F[22,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[4,x]
F[22,x] = F[14,x]+F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[26,x]+F[27,x]
F[26,x] = F[19,x]
F[27,x] = F[23,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[1,x]+F[4,x]
F[33,x] = F[2,x]+F[34,x]
F[34,x] = F[15,x]+F[35,x]+F[92,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,x]+F[46,x]
F[37,x] = F[38,x]+F[7,x]
F[38,x] = F[15,x]+F[16,x]+F[39,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]
F[41,x] = F[30,x]+F[42,x]
F[42,x] = F[15,x]+F[28,x]+F[43,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[11,x]+F[45,x]
F[45,x] = F[20,x]
F[46,x] = F[34,x]+F[47,x]
F[47,x] = F[15,x]+F[48,x]+F[80,x]+F[91,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[50,x]+F[59,x]
F[50,x] = F[38,x]+F[51,x]
F[51,x] = F[15,x]+F[24,x]+F[52,x]+F[58,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[54,x]
F[54,x] = F[42,x]+F[55,x]
F[55,x] = F[56,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[19,x]
F[58,x] = 0
F[59,x] = F[47,x]+F[60,x]
F[60,x] = F[15,x]+F[61,x]+F[73,x]+F[78,x]+F[79,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[63,x]+F[67,x]
F[63,x] = F[51,x]+F[64,x]
F[64,x] = F[65,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[55,x]
F[67,x] = F[60,x]+F[68,x]
F[68,x] = F[69,x]
F[69,x] = F[4,x]*F[70,x]
F[70,x] = F[71,x]+F[72,x]
F[71,x] = F[64,x]
F[72,x] = F[68,x]
F[73,x] = F[4,x]*F[74,x]
F[74,x] = F[75,x]
F[75,x] = F[76,x]
F[76,x] = F[4,x]*F[77,x]
F[77,x] = F[19,x]+F[55,x]
F[78,x] = 0
F[79,x] = 0
F[80,x] = F[4,x]*F[81,x]
F[81,x] = F[82,x]
F[82,x] = F[75,x]+F[83,x]
F[83,x] = F[15,x]+F[84,x]+F[86,x]
F[84,x] = F[4,x]*F[85,x]
F[85,x] = F[11,x]+F[42,x]
F[86,x] = F[4,x]*F[87,x]
F[87,x] = F[88,x]
F[88,x] = F[89,x]
F[89,x] = F[4,x]*F[90,x]
F[90,x] = F[1,x]+F[30,x]
F[91,x] = 0
F[92,x] = F[4,x]*F[93,x]
F[93,x] = F[94,x]
F[94,x] = F[83,x]+F[88,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_{33}\! \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_{28}\! \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_{22}\! \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_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{14}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{27}\! \left(x \right)
F_{26}\! \left(x \right) = F_{19}\! \left(x \right)
F_{27}\! \left(x \right) = F_{23}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{2}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{15}\! \left(x \right)+F_{35}\! \left(x \right)+F_{92}\! \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_{46}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{7}\! \left(x \right)
F_{38}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{30}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{15}\! \left(x \right)+F_{28}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{11}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{20}\! \left(x \right)
F_{46}\! \left(x \right) = F_{34}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{15}\! \left(x \right)+F_{48}\! \left(x \right)+F_{80}\! \left(x \right)+F_{91}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{59}\! \left(x \right)
F_{50}\! \left(x \right) = F_{38}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{15}\! \left(x \right)+F_{24}\! \left(x \right)+F_{52}\! \left(x \right)+F_{58}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{42}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{19}\! \left(x \right)
F_{58}\! \left(x \right) = 0
F_{59}\! \left(x \right) = F_{47}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{15}\! \left(x \right)+F_{61}\! \left(x \right)+F_{73}\! \left(x \right)+F_{78}\! \left(x \right)+F_{79}\! \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_{67}\! \left(x \right)
F_{63}\! \left(x \right) = F_{51}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{55}\! \left(x \right)
F_{67}\! \left(x \right) = F_{60}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{4}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)+F_{72}\! \left(x \right)
F_{71}\! \left(x \right) = F_{64}\! \left(x \right)
F_{72}\! \left(x \right) = F_{68}\! \left(x \right)
F_{73}\! \left(x \right) = F_{4}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{4}\! \left(x \right) F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{19}\! \left(x \right)+F_{55}\! \left(x \right)
F_{78}\! \left(x \right) = 0
F_{79}\! \left(x \right) = 0
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_{82}\! \left(x \right) = F_{75}\! \left(x \right)+F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{15}\! \left(x \right)+F_{84}\! \left(x \right)+F_{86}\! \left(x \right)
F_{84}\! \left(x \right) = F_{4}\! \left(x \right) F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{11}\! \left(x \right)+F_{42}\! \left(x \right)
F_{86}\! \left(x \right) = F_{4}\! \left(x \right) F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = F_{4}\! \left(x \right) F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = F_{1}\! \left(x \right)+F_{30}\! \left(x \right)
F_{91}\! \left(x \right) = 0
F_{92}\! \left(x \right) = F_{4}\! \left(x \right) F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{94}\! \left(x \right)
F_{94}\! \left(x \right) = F_{83}\! \left(x \right)+F_{88}\! \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_33(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_28(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_22(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_4(x))
Eq(F_22(x), F_14(x) + F_23(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_26(x) + F_27(x))
Eq(F_26(x), F_19(x))
Eq(F_27(x), F_23(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_1(x) + F_4(x))
Eq(F_33(x), F_2(x) + F_34(x))
Eq(F_34(x), F_15(x) + F_35(x) + F_92(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_37(x) + F_46(x))
Eq(F_37(x), F_38(x) + F_7(x))
Eq(F_38(x), F_15(x) + F_16(x) + F_39(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_30(x) + F_42(x))
Eq(F_42(x), F_15(x) + F_28(x) + F_43(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_11(x) + F_45(x))
Eq(F_45(x), F_20(x))
Eq(F_46(x), F_34(x) + F_47(x))
Eq(F_47(x), F_15(x) + F_48(x) + F_80(x) + F_91(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_59(x))
Eq(F_50(x), F_38(x) + F_51(x))
Eq(F_51(x), F_15(x) + F_24(x) + F_52(x) + F_58(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_42(x) + F_55(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_19(x))
Eq(F_58(x), 0)
Eq(F_59(x), F_47(x) + F_60(x))
Eq(F_60(x), F_15(x) + F_61(x) + F_73(x) + F_78(x) + F_79(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_63(x) + F_67(x))
Eq(F_63(x), F_51(x) + F_64(x))
Eq(F_64(x), F_65(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_55(x))
Eq(F_67(x), F_60(x) + F_68(x))
Eq(F_68(x), F_69(x))
Eq(F_69(x), F_4(x)*F_70(x))
Eq(F_70(x), F_71(x) + F_72(x))
Eq(F_71(x), F_64(x))
Eq(F_72(x), F_68(x))
Eq(F_73(x), F_4(x)*F_74(x))
Eq(F_74(x), F_75(x))
Eq(F_75(x), F_76(x))
Eq(F_76(x), F_4(x)*F_77(x))
Eq(F_77(x), F_19(x) + F_55(x))
Eq(F_78(x), 0)
Eq(F_79(x), 0)
Eq(F_80(x), F_4(x)*F_81(x))
Eq(F_81(x), F_82(x))
Eq(F_82(x), F_75(x) + F_83(x))
Eq(F_83(x), F_15(x) + F_84(x) + F_86(x))
Eq(F_84(x), F_4(x)*F_85(x))
Eq(F_85(x), F_11(x) + F_42(x))
Eq(F_86(x), F_4(x)*F_87(x))
Eq(F_87(x), F_88(x))
Eq(F_88(x), F_89(x))
Eq(F_89(x), F_4(x)*F_90(x))
Eq(F_90(x), F_1(x) + F_30(x))
Eq(F_91(x), 0)
Eq(F_92(x), F_4(x)*F_93(x))
Eq(F_93(x), F_94(x))
Eq(F_94(x), F_83(x) + F_88(x))
Pack JSON:
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Specification JSON:
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