0132_0213_1032_2130_2301

Counting sequence:
1, 1, 2, 6, 19, 50, 114, 236, 459, 860, 1582, 2898, 5339, 9950, 18802, 36016, 69811, 136600, 269182, 533126, 1059539, 2110602, 4210642, 8408276, 16800699, 33582260, 67141614, 134256026, 268479979, 536922390, 1073801042, 2147551448, 4295044579, 8590022320, 17179968382, 34359850126, 68719602211, 137439093890, 274878063602, 549755988156, 1099511821099, 2199023469452, 4398046747182, 8796093282146, 17592186329979, 35184372401870, 70368744520114, 140737488729216, 281474977118099, 562949953864520, 1125899907323902, 2251799814206998, 4503599627935219, 9007199255351290, 18014398510140562, 36028797019673636, 72057594038691611, 144115188076676580, 288230376152592622, 576460752304367786, 1152921504607858059, 2305843009214775302, 4611686018428543122, 9223372036856008616, 18446744073710865859, 36893488147420502880, 73786976294839695614, 147573952589677995806, 295147905179354506819, 590295810358707435250, 1180591620717413194162, 2361183241434824609548, 4722366482869647333259, 9444732965739292668860, 18889465931478583223342, 37778931862957164210546, 75557863725914326058011, 151115727451828649620670, 302231454903657296608242, 604462909807314590440016, 1208925819614629177954419, 2417851639229258352828152, 4835703278458516702414462, 9671406556917033401419686, 19342813113834066799256339, 38685626227668133594749290, 77371252455336267185548114, 154742504910672534366951796, 309485009821345068729558139, 618970019642690137454562580, 1237940039285380274904355822, 2475880078570760549803719098, 4951760157141521099602214699, 9903520314283042199198967030, 19807040628566084398392224722, 39614081257132168796778484856, 79228162514264337593550741411, 158456325028528675187094982160, 316912650057057350374183182462, 633825300114114700748359292846, 1267650600228229401496711214179

Generating function in Maple syntax:
-(2*x^7-5*x^6+9*x^5-12*x^4+18*x^3-15*x^2+6*x-1)/(2*x-1)/(x-1)^5

Generating function in latex syntax:
-\frac{2 x^{7}-5 x^{6}+9 x^{5}-12 x^{4}+18 x^{3}-15 x^{2}+6 x -1}{\left(2 x -1\right) \left(x -1\right)^{5}}

Generating function in sympy syntax:
(-2*x**7 + 5*x**6 - 9*x**5 + 12*x**4 - 18*x**3 + 15*x**2 - 6*x + 1)/((x - 1)**5*(2*x - 1))

Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x-1)^5*F(x)+2*x^7-5*x^6+9*x^5-12*x^4+18*x^3-15*x^2+6*x-1 = 0

Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x -1\right)^{5} F \! \left(x \right)+2 x^{7}-5 x^{6}+9 x^{5}-12 x^{4}+18 x^{3}-15 x^{2}+6 x -1 = 0

Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,3+2^n-11/3*n+11/12*n^2-1/3*n^3+1/12*n^4)

Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 3+2^{n}-\frac{11 n}{3}+\frac{11 n^{2}}{12}-\frac{n^{3}}{3}+\frac{n^{4}}{12} & \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) = 114
a(7) = 236
a(n+1) = -1/12*n^4+2/3*n^3-17/12*n^2+2*a(n)+29/6*n-6, n >= 8

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) = 114
a \! \left(7\right) = 236
a \! \left(n +1\right) = -\frac{n^{4}}{12}+\frac{2 n^{3}}{3}-\frac{17 n^{2}}{12}+2 a \! \left(n \right)+\frac{29 n}{6}-6, \quad n \geq 8

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18362/
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[12,x]*F[4,x]
F[4,x] = F[21,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[6,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[17,x]+F[20,x]
F[17,x] = F[18,x]
F[18,x] = F[12,x]*F[19,x]
F[19,x] = F[1,x]+F[17,x]
F[20,x] = F[15,x]
F[21,x] = F[22,x]+F[38,x]
F[22,x] = F[23,x]
F[23,x] = F[12,x]*F[24,x]
F[24,x] = F[25,x]+F[5,x]
F[25,x] = F[22,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[12,x]*F[28,x]
F[28,x] = F[16,x]+F[29,x]
F[29,x] = F[30,x]+F[31,x]
F[30,x] = F[27,x]
F[31,x] = F[32,x]
F[32,x] = F[12,x]*F[33,x]
F[33,x] = F[34,x]+F[37,x]
F[34,x] = F[35,x]
F[35,x] = F[12,x]*F[36,x]
F[36,x] = F[17,x]+F[34,x]
F[37,x] = F[32,x]
F[38,x] = F[39,x]+F[40,x]+F[76,x]
F[39,x] = 0
F[40,x] = F[12,x]*F[41,x]
F[41,x] = F[42,x]+F[54,x]
F[42,x] = F[10,x]+F[43,x]
F[43,x] = F[15,x]+F[39,x]+F[44,x]
F[44,x] = F[12,x]*F[45,x]
F[45,x] = F[46,x]+F[49,x]
F[46,x] = F[10,x]+F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[12,x]*F[46,x]
F[49,x] = F[14,x]+F[50,x]
F[50,x] = F[51,x]
F[51,x] = F[12,x]*F[52,x]
F[52,x] = F[20,x]+F[53,x]
F[53,x] = F[51,x]
F[54,x] = F[55,x]+F[61,x]
F[55,x] = F[39,x]+F[40,x]+F[56,x]
F[56,x] = F[12,x]*F[57,x]
F[57,x] = F[17,x]+F[58,x]
F[58,x] = F[39,x]+F[56,x]+F[59,x]
F[59,x] = F[12,x]*F[60,x]
F[60,x] = F[10,x]+F[58,x]
F[61,x] = 2*F[39,x]+F[62,x]+F[69,x]
F[62,x] = F[12,x]*F[63,x]
F[63,x] = F[64,x]+F[67,x]
F[64,x] = F[58,x]+F[65,x]
F[65,x] = F[66,x]
F[66,x] = F[12,x]*F[64,x]
F[67,x] = F[68,x]+F[72,x]
F[68,x] = F[69,x]
F[69,x] = F[12,x]*F[70,x]
F[70,x] = F[34,x]+F[71,x]
F[71,x] = F[69,x]
F[72,x] = F[73,x]
F[73,x] = F[12,x]*F[74,x]
F[74,x] = F[71,x]+F[75,x]
F[75,x] = F[73,x]
F[76,x] = F[12,x]*F[77,x]
F[77,x] = F[57,x]+F[78,x]
F[78,x] = F[79,x]+F[86,x]
F[79,x] = F[39,x]+F[80,x]+F[82,x]
F[80,x] = F[12,x]*F[81,x]
F[81,x] = F[10,x]+F[20,x]
F[82,x] = F[12,x]*F[83,x]
F[83,x] = F[16,x]+F[84,x]
F[84,x] = F[79,x]+F[85,x]
F[85,x] = F[32,x]
F[86,x] = F[69,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{21}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{6}\! \left(x \right)
F_{14}\! \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_{20}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \left(x \right)
F_{20}\! \left(x \right) = F_{15}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{38}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{5}\! \left(x \right)
F_{25}\! \left(x \right) = F_{22}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{16}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{31}\! \left(x \right)
F_{30}\! \left(x \right) = F_{27}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{12}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{37}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{12}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{17}\! \left(x \right)+F_{34}\! \left(x \right)
F_{37}\! \left(x \right) = F_{32}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{40}\! \left(x \right)+F_{76}\! \left(x \right)
F_{39}\! \left(x \right) = 0
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{54}\! \left(x \right)
F_{42}\! \left(x \right) = F_{10}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{15}\! \left(x \right)+F_{39}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \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_{10}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{46}\! \left(x \right)
F_{49}\! \left(x \right) = F_{14}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{20}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{51}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{61}\! \left(x \right)
F_{55}\! \left(x \right) = F_{39}\! \left(x \right)+F_{40}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{12}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{17}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{39}\! \left(x \right)+F_{56}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{12}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{10}\! \left(x \right)+F_{58}\! \left(x \right)
F_{61}\! \left(x \right) = 2 F_{39}\! \left(x \right)+F_{62}\! \left(x \right)+F_{69}\! \left(x \right)
F_{62}\! \left(x \right) = F_{12}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{67}\! \left(x \right)
F_{64}\! \left(x \right) = F_{58}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{12}\! \left(x \right) F_{64}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)+F_{72}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{12}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{34}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{69}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{12}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{71}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{73}\! \left(x \right)
F_{76}\! \left(x \right) = F_{12}\! \left(x \right) F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{57}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{79}\! \left(x \right)+F_{86}\! \left(x \right)
F_{79}\! \left(x \right) = F_{39}\! \left(x \right)+F_{80}\! \left(x \right)+F_{82}\! \left(x \right)
F_{80}\! \left(x \right) = F_{12}\! \left(x \right) F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{10}\! \left(x \right)+F_{20}\! \left(x \right)
F_{82}\! \left(x \right) = F_{12}\! \left(x \right) F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{16}\! \left(x \right)+F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{79}\! \left(x \right)+F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{32}\! \left(x \right)
F_{86}\! \left(x \right) = F_{69}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_21(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_6(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_17(x) + F_20(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_12(x)*F_19(x))
Eq(F_19(x), F_1(x) + F_17(x))
Eq(F_20(x), F_15(x))
Eq(F_21(x), F_22(x) + F_38(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_12(x)*F_24(x))
Eq(F_24(x), F_25(x) + F_5(x))
Eq(F_25(x), F_22(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_12(x)*F_28(x))
Eq(F_28(x), F_16(x) + F_29(x))
Eq(F_29(x), F_30(x) + F_31(x))
Eq(F_30(x), F_27(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_12(x)*F_33(x))
Eq(F_33(x), F_34(x) + F_37(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_12(x)*F_36(x))
Eq(F_36(x), F_17(x) + F_34(x))
Eq(F_37(x), F_32(x))
Eq(F_38(x), F_39(x) + F_40(x) + F_76(x))
Eq(F_39(x), 0)
Eq(F_40(x), F_12(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_54(x))
Eq(F_42(x), F_10(x) + F_43(x))
Eq(F_43(x), F_15(x) + F_39(x) + F_44(x))
Eq(F_44(x), F_12(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_49(x))
Eq(F_46(x), F_10(x) + F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_12(x)*F_46(x))
Eq(F_49(x), F_14(x) + F_50(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_12(x)*F_52(x))
Eq(F_52(x), F_20(x) + F_53(x))
Eq(F_53(x), F_51(x))
Eq(F_54(x), F_55(x) + F_61(x))
Eq(F_55(x), F_39(x) + F_40(x) + F_56(x))
Eq(F_56(x), F_12(x)*F_57(x))
Eq(F_57(x), F_17(x) + F_58(x))
Eq(F_58(x), F_39(x) + F_56(x) + F_59(x))
Eq(F_59(x), F_12(x)*F_60(x))
Eq(F_60(x), F_10(x) + F_58(x))
Eq(F_61(x), 2*F_39(x) + F_62(x) + F_69(x))
Eq(F_62(x), F_12(x)*F_63(x))
Eq(F_63(x), F_64(x) + F_67(x))
Eq(F_64(x), F_58(x) + F_65(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_12(x)*F_64(x))
Eq(F_67(x), F_68(x) + F_72(x))
Eq(F_68(x), F_69(x))
Eq(F_69(x), F_12(x)*F_70(x))
Eq(F_70(x), F_34(x) + F_71(x))
Eq(F_71(x), F_69(x))
Eq(F_72(x), F_73(x))
Eq(F_73(x), F_12(x)*F_74(x))
Eq(F_74(x), F_71(x) + F_75(x))
Eq(F_75(x), F_73(x))
Eq(F_76(x), F_12(x)*F_77(x))
Eq(F_77(x), F_57(x) + F_78(x))
Eq(F_78(x), F_79(x) + F_86(x))
Eq(F_79(x), F_39(x) + F_80(x) + F_82(x))
Eq(F_80(x), F_12(x)*F_81(x))
Eq(F_81(x), F_10(x) + F_20(x))
Eq(F_82(x), F_12(x)*F_83(x))
Eq(F_83(x), F_16(x) + F_84(x))
Eq(F_84(x), F_79(x) + F_85(x))
Eq(F_85(x), F_32(x))
Eq(F_86(x), F_69(x))
Pack JSON:
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Specification JSON:
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