0132_1032_1320_2103_3021
Counting sequence:
1, 1, 2, 6, 19, 54, 138, 333, 788, 1867, 4464, 10779, 26231, 64175, 157533, 387472, 954112, 2350861, 5794249, 14283713, 35214572, 86820504, 214057934, 527770126, 1301249042, 3208315482, 7910322625, 19503455070, 48087147715, 118562276691, 292323725826, 720744954468, 1777048009259, 4381438431508, 10802748531282, 26634945982227, 65670356552516, 161914941876809, 399212822673816, 984287650949593, 2426831316038791, 5983525477405861, 14752808282253517, 36374099690074310, 89682933781218224, 221119661521230579, 545186276249020385, 1344195598733547031, 3314209998252738980, 8171420828090240166, 20147220117297862174, 49674406323498417060, 122475787191775026922, 301972777501501554624, 744535393017626277297, 1835705046138322355436, 4526061551969662633955, 11159327156239917521425, 27514116003526323909042, 67838012888814695853474, 167259453006346916351123, 412390095591893157777842, 1016777156002282136825034, 2506936505068703507368249, 6181030527039082818580228, 15239771050819660147848391, 37574741083289337020485360, 92643200660176295798322743, 228418410376717478534286119, 563181861455856446206857627, 1388564995920367279550832749, 3423605907532339322771759452, 8441144234895090231953298608, 20812242389679766152606740313, 51314066106840631418975584489, 126518485183648965608728362957, 311940337369432003797855573676, 769111121880005081404707049140, 1896298256223807203455735224990, 4675458427603499418192972175194, 11527675794934976899707333514898, 28422305810393832118231678641766, 70077219549710942228968963292625, 172780376531679539023839460271898, 426002325809344941305208060306131, 1050339079228114896788541829489871, 2589685817460773497699645396981842, 6385054851130554330771496621123840, 15742807554902694524953384987773067, 38815013416340326939054993036125520, 95701180444239498123767537068704562, 235958077359964441374655338057195263, 581771447467681255696948645993309076, 1434398944403621310285820504893036549, 3536612772355972286782504803776837816, 8719770709808826362791762628621735029, 21499215810666307948111988010281487767, 53007848010689016838836653309347403985, 130694625118851006703161961071846984061, 322236907853957866462800541771494968466, 794498049853643482851902321654786741056
Generating function in Maple syntax:
(2*x-1)*(2*x^7-3*x^5+9*x^4-12*x^3+11*x^2-5*x+1)/(3*x^3-5*x^2+4*x-1)/(x-1)^4
Generating function in latex syntax:
\frac{\left(2 x -1\right) \left(2 x^{7}-3 x^{5}+9 x^{4}-12 x^{3}+11 x^{2}-5 x +1\right)}{\left(3 x^{3}-5 x^{2}+4 x -1\right) \left(x -1\right)^{4}}
Generating function in sympy syntax:
(2*x - 1)*(2*x**7 - 3*x**5 + 9*x**4 - 12*x**3 + 11*x**2 - 5*x + 1)/((x - 1)**4*(3*x**3 - 5*x**2 + 4*x - 1))
Implicit equation for the generating function in Maple syntax:
-(3*x^3-5*x^2+4*x-1)*(x-1)^4*F(x)+(2*x-1)*(2*x^7-3*x^5+9*x^4-12*x^3+11*x^2-5*x+1) = 0
Implicit equation for the generating function in latex syntax:
-\left(3 x^{3}-5 x^{2}+4 x -1\right) \left(x -1\right)^{4} F \! \left(x \right)+\left(2 x -1\right) \left(2 x^{7}-3 x^{5}+9 x^{4}-12 x^{3}+11 x^{2}-5 x +1\right) = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,n = 1,1,1/810216*(-1550*2^(1/3)*((I-48/775*31^(1/2))*3^(1/2)-144/775*I*31^(1/2)+1)*(47+9*31^(1/2)*3^(1/2))^(2/3)+300080+341*((I-39/31*31^(1/2))*3^(1/2)+117/31*I*31^(1/2)-1)*2^(2/3)*(47+9*31^(1/2)*3^(1/2))^(1/3))*(47/8712*2^(1/3)*((I-9/47*31^(1/2))*3^(1/2)-27/47*I*31^(1/2)+1)*(47+9*31^(1/2)*3^(1/2))^(2/3)-1/36*I*3^(1/2)*(188+36*31^(1/2)*3^(1/2))^(1/3)+1/36*(188+36*31^(1/2)*3^(1/2))^(1/3)+5/9)^(-n)+1/810216*(-341*2^(2/3)*((I+39/31*31^(1/2))*3^(1/2)+117/31*I*31^(1/2)+1)*(47+9*31^(1/2)*3^(1/2))^(1/3)+300080+1550*2^(1/3)*((I+48/775*31^(1/2))*3^(1/2)-144/775*I*31^(1/2)-1)*(47+9*31^(1/2)*3^(1/2))^(2/3))*(-47/8712*2^(1/3)*((I+9/47*31^(1/2))*3^(1/2)-27/47*I*31^(1/2)-1)*(47+9*31^(1/2)*3^(1/2))^(2/3)+1/36*I*3^(1/2)*(188+36*31^(1/2)*3^(1/2))^(1/3)+1/36*(188+36*31^(1/2)*3^(1/2))^(1/3)+5/9)^(-n)+1/810216*((858*2^(2/3)*31^(1/2)*3^(1/2)+682*2^(2/3))*(47+9*31^(1/2)*3^(1/2))^(1/3)+300080+(-192*31^(1/2)*3^(1/2)*2^(1/3)+3100*2^(1/3))*(47+9*31^(1/2)*3^(1/2))^(2/3))*(1/4356*2^(1/3)*(9*31^(1/2)*3^(1/2)-47)*(47+9*31^(1/2)*3^(1/2))^(2/3)-1/18*(188+36*31^(1/2)*3^(1/2))^(1/3)+5/9)^(-n)+1/2*n^3-7/2*n^2+8*n-7)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ \frac{\left(-1550 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}-\frac{48 \sqrt{31}}{775}\right) \sqrt{3}-\frac{144 \,\mathrm{I} \sqrt{31}}{775}+1\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}+300080+341 \left(\left(\mathrm{I}-\frac{39 \sqrt{31}}{31}\right) \sqrt{3}+\frac{117 \,\mathrm{I} \sqrt{31}}{31}-1\right) 2^{\frac{2}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{47 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}-\frac{9 \sqrt{31}}{47}\right) \sqrt{3}-\frac{27 \,\mathrm{I} \sqrt{31}}{47}+1\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{8712}-\frac{\mathrm{I} \sqrt{3}\, \left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{\left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{5}{9}\right)^{-n}}{810216}\\+\\\frac{\left(-341 \,2^{\frac{2}{3}} \left(\left(\mathrm{I}+\frac{39 \sqrt{31}}{31}\right) \sqrt{3}+\frac{117 \,\mathrm{I} \sqrt{31}}{31}+1\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}+300080+1550 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}+\frac{48 \sqrt{31}}{775}\right) \sqrt{3}-\frac{144 \,\mathrm{I} \sqrt{31}}{775}-1\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(-\frac{47 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}+\frac{9 \sqrt{31}}{47}\right) \sqrt{3}-\frac{27 \,\mathrm{I} \sqrt{31}}{47}-1\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{8712}+\frac{\mathrm{I} \sqrt{3}\, \left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{\left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{5}{9}\right)^{-n}}{810216}\\+\\\frac{\left(\left(858 \,2^{\frac{2}{3}} \sqrt{31}\, \sqrt{3}+682 \,2^{\frac{2}{3}}\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}+300080+\left(-192 \sqrt{31}\, \sqrt{3}\, 2^{\frac{1}{3}}+3100 \,2^{\frac{1}{3}}\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{2^{\frac{1}{3}} \left(9 \sqrt{31}\, \sqrt{3}-47\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{4356}-\frac{\left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{18}+\frac{5}{9}\right)^{-n}}{810216}\\+\frac{n^{3}}{2}-\frac{7 n^{2}}{2}+8 n -7 & \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) = 54
a(6) = 138
a(7) = 333
a(8) = 788
a(n+3) = -1/2*n^3+7/2*n^2+3*a(n)-5*a(n+1)+4*a(n+2)-11*n+14, 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) = 19
a \! \left(5\right) = 54
a \! \left(6\right) = 138
a \! \left(7\right) = 333
a \! \left(8\right) = 788
a \! \left(n +3\right) = -\frac{n^{3}}{2}+\frac{7 n^{2}}{2}+3 a \! \left(n \right)-5 a \! \left(n +1\right)+4 a \! \left(n +2\right)-11 n +14, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19797/
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[28,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[27,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[18,x]+F[21,x]
F[18,x] = F[11,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[18,x]*F[4,x]
F[21,x] = F[22,x]+F[25,x]
F[22,x] = F[23,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[22,x]+F[7,x]
F[25,x] = F[26,x]
F[26,x] = F[21,x]*F[4,x]
F[27,x] = F[13,x]*F[4,x]
F[28,x] = F[29,x]+F[7,x]
F[29,x] = F[15,x]+F[30,x]+F[44,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[32,x]+F[36,x]
F[32,x] = F[11,x]+F[33,x]
F[33,x] = F[15,x]+F[20,x]+F[34,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[11,x]+F[33,x]
F[36,x] = F[22,x]+F[37,x]
F[37,x] = 2*F[15,x]+F[26,x]+F[38,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]+F[43,x]
F[40,x] = F[41,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[18,x]
F[43,x] = F[38,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]+F[49,x]
F[46,x] = F[47,x]+F[7,x]
F[47,x] = F[15,x]+F[30,x]+F[48,x]
F[48,x] = F[4,x]*F[46,x]
F[49,x] = F[50,x]+F[51,x]
F[50,x] = F[41,x]
F[51,x] = 2*F[15,x]+F[52,x]+F[59,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[54,x]
F[54,x] = F[19,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]
F[58,x] = F[56,x]
F[59,x] = F[4,x]*F[49,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_{28}\! \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_{27}\! \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_{21}\! \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_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{25}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{22}\! \left(x \right)+F_{7}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{7}\! \left(x \right)
F_{29}\! \left(x \right) = F_{15}\! \left(x \right)+F_{30}\! \left(x \right)+F_{44}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{36}\! \left(x \right)
F_{32}\! \left(x \right) = F_{11}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{15}\! \left(x \right)+F_{20}\! \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_{11}\! \left(x \right)+F_{33}\! \left(x \right)
F_{36}\! \left(x \right) = F_{22}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{26}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{43}\! \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_{18}\! \left(x \right)
F_{43}\! \left(x \right) = F_{38}\! \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_{49}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{7}\! \left(x \right)
F_{47}\! \left(x \right) = F_{15}\! \left(x \right)+F_{30}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{51}\! \left(x \right)
F_{50}\! \left(x \right) = F_{41}\! \left(x \right)
F_{51}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{52}\! \left(x \right)+F_{59}\! \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_{19}\! \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)
F_{58}\! \left(x \right) = F_{56}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \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_28(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_27(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_21(x))
Eq(F_18(x), F_11(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_18(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_25(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_22(x) + F_7(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_21(x)*F_4(x))
Eq(F_27(x), F_13(x)*F_4(x))
Eq(F_28(x), F_29(x) + F_7(x))
Eq(F_29(x), F_15(x) + F_30(x) + F_44(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_32(x) + F_36(x))
Eq(F_32(x), F_11(x) + F_33(x))
Eq(F_33(x), F_15(x) + F_20(x) + F_34(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_11(x) + F_33(x))
Eq(F_36(x), F_22(x) + F_37(x))
Eq(F_37(x), 2*F_15(x) + F_26(x) + F_38(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_40(x) + F_43(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_18(x))
Eq(F_43(x), F_38(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_49(x))
Eq(F_46(x), F_47(x) + F_7(x))
Eq(F_47(x), F_15(x) + F_30(x) + F_48(x))
Eq(F_48(x), F_4(x)*F_46(x))
Eq(F_49(x), F_50(x) + F_51(x))
Eq(F_50(x), F_41(x))
Eq(F_51(x), 2*F_15(x) + F_52(x) + F_59(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_19(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) + F_58(x))
Eq(F_58(x), F_56(x))
Eq(F_59(x), F_4(x)*F_49(x))
Pack JSON:
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Specification JSON:
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