0231_1230_2013_3021

Counting sequence:
1, 1, 2, 6, 20, 61, 166, 412, 956, 2115, 4526, 9466, 19492, 39721, 80390, 161976, 325436, 652687, 1307566, 2617750, 5238596, 10480821, 20965862, 41936596, 83878780, 167763931, 335535086, 671078322, 1342165796, 2684341825, 5368695046, 10737402736, 21474819452, 42949654311, 85899325550, 171798669646, 343597359556, 687194741197, 1374389506406, 2748779038860, 5497558105916, 10995116242291, 21990232517422, 43980465070186, 87960930178340, 175921860397401, 351843720838406, 703687441723432, 1407374883496636, 2814749767046335, 5629499534149166, 11258999068358406, 22517998136780612, 45035996273628901, 90071992547329510, 180143985094734916, 360287970189550076, 720575940379184907, 1441151880758459246, 2882303761517012770, 5764607523034124836, 11529215046068354161, 23058430092136818182, 46116860184273751776, 92233720368547624700, 184467440737095376471, 368934881474190886126, 737869762948381911742, 1475739525896763969476, 2951479051793528091645, 5902958103587056342886, 11805916207174112852476, 23611832414348225878972, 47223664828696451939491, 94447329657392904068270, 188894659314785808333786, 377789318629571616872996, 755578637259143233959817, 1511157274518286468142086, 3022314549036572936515480, 6044629098073145873271356, 12089258196146291746792431, 24178516392292583493844142, 48357032784585166987957366, 96714065569170333976193860, 193428131138340667952677141, 386856262276681335905654246, 773712524553362671811619252, 1547425049106725343623560316, 3094850098213450687247453755, 6189700196426901374495252206, 12379400392853802748990860946, 24758800785707605497982090532, 49517601571415210995964562081, 99035203142830421991929517830, 198070406285660843983859442256, 396140812571321687967719304316, 792281625142643375935439041927, 1584563250285286751870878530926, 3169126500570573503741757522990, 6338253001141147007483515521476

Generating function in Maple syntax:
-(5*x^4-8*x^3+10*x^2-5*x+1)/(2*x-1)/(x-1)^4

Generating function in latex syntax:
-\frac{5 x^{4}-8 x^{3}+10 x^{2}-5 x +1}{\left(2 x -1\right) \left(x -1\right)^{4}}

Generating function in sympy syntax:
(-5*x**4 + 8*x**3 - 10*x**2 + 5*x - 1)/((x - 1)**4*(2*x - 1))

Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x-1)^4*F(x)+5*x^4-8*x^3+10*x^2-5*x+1 = 0

Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x -1\right)^{4} F \! \left(x \right)+5 x^{4}-8 x^{3}+10 x^{2}-5 x +1 = 0

Explicit closed form in Maple syntax:
5*2^n-4*n-4-1/2*n^2-1/2*n^3

Explicit closed form in latex syntax:
5 \,2^{n}-4 n -4-\frac{n^{2}}{2}-\frac{n^{3}}{2}

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(n+1) = 2*a(n)+1/2*(n-1)*(n^2-n+2), n >= 5

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(n +1\right) = 2 a \! \left(n \right)+\frac{\left(n -1\right) \left(n^{2}-n +2\right)}{2}, \quad n \geq 5

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15901/
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[6,x]
F[10,x] = F[11,x]+F[23,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[14,x]+F[17,x]
F[14,x] = F[1,x]+F[15,x]
F[15,x] = F[16,x]
F[16,x] = F[14,x]*F[4,x]
F[17,x] = F[11,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[22,x]
F[21,x] = F[15,x]
F[22,x] = F[18,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[7,x]
F[27,x] = F[23,x]
F[28,x] = F[29,x]+F[44,x]
F[29,x] = F[30,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[14,x]+F[32,x]
F[32,x] = F[29,x]+F[33,x]
F[33,x] = F[34,x]+F[35,x]+F[38,x]
F[34,x] = 0
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[21,x]+F[37,x]
F[37,x] = F[33,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[29,x]+F[40,x]
F[40,x] = F[41,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[21,x]+F[43,x]
F[43,x] = F[40,x]
F[44,x] = F[34,x]+F[45,x]+F[71,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]+F[57,x]
F[47,x] = F[48,x]+F[7,x]
F[48,x] = F[49,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[51,x]+F[52,x]
F[51,x] = F[15,x]
F[52,x] = F[53,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[55,x]+F[56,x]
F[55,x] = F[15,x]
F[56,x] = F[53,x]
F[57,x] = F[44,x]+F[58,x]
F[58,x] = 2*F[34,x]+F[59,x]+F[63,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[61,x]+F[62,x]
F[61,x] = F[48,x]
F[62,x] = F[58,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[65,x]+F[66,x]
F[65,x] = F[33,x]
F[66,x] = F[67,x]
F[67,x] = F[4,x]*F[68,x]
F[68,x] = F[69,x]+F[70,x]
F[69,x] = F[33,x]
F[70,x] = F[67,x]
F[71,x] = F[4,x]*F[72,x]
F[72,x] = F[73,x]+F[78,x]
F[73,x] = F[29,x]+F[74,x]
F[74,x] = F[75,x]
F[75,x] = F[4,x]*F[76,x]
F[76,x] = F[26,x]+F[77,x]
F[77,x] = F[74,x]
F[78,x] = F[79,x]+F[88,x]
F[79,x] = F[34,x]+F[35,x]+F[80,x]
F[80,x] = F[4,x]*F[81,x]
F[81,x] = F[39,x]+F[82,x]
F[82,x] = F[79,x]+F[83,x]
F[83,x] = F[84,x]
F[84,x] = F[4,x]*F[85,x]
F[85,x] = F[86,x]+F[87,x]
F[86,x] = F[40,x]
F[87,x] = F[83,x]
F[88,x] = F[89,x]
F[89,x] = F[4,x]*F[90,x]
F[90,x] = F[91,x]+F[92,x]
F[91,x] = F[74,x]
F[92,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_{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_{6}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{23}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{17}\! \left(x \right)
F_{14}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{11}\! \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_{22}\! \left(x \right)
F_{21}\! \left(x \right) = F_{15}\! \left(x \right)
F_{22}\! \left(x \right) = F_{18}\! \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_{7}\! \left(x \right)
F_{27}\! \left(x \right) = F_{23}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{44}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{14}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{29}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{35}\! \left(x \right)+F_{38}\! \left(x \right)
F_{34}\! \left(x \right) = 0
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{21}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{33}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{29}\! \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_{21}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{40}\! \left(x \right)
F_{44}\! \left(x \right) = F_{34}\! \left(x \right)+F_{45}\! \left(x \right)+F_{71}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{57}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{7}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{52}\! \left(x \right)
F_{51}\! \left(x \right) = F_{15}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{56}\! \left(x \right)
F_{55}\! \left(x \right) = F_{15}\! \left(x \right)
F_{56}\! \left(x \right) = F_{53}\! \left(x \right)
F_{57}\! \left(x \right) = F_{44}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = 2 F_{34}\! \left(x \right)+F_{59}\! \left(x \right)+F_{63}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)+F_{62}\! \left(x \right)
F_{61}\! \left(x \right) = F_{48}\! \left(x \right)
F_{62}\! \left(x \right) = F_{58}\! \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_{66}\! \left(x \right)
F_{65}\! \left(x \right) = F_{33}\! \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_{69}\! \left(x \right)+F_{70}\! \left(x \right)
F_{69}\! \left(x \right) = F_{33}\! \left(x \right)
F_{70}\! \left(x \right) = F_{67}\! \left(x \right)
F_{71}\! \left(x \right) = F_{4}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)+F_{78}\! \left(x \right)
F_{73}\! \left(x \right) = F_{29}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{4}\! \left(x \right) F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{26}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{74}\! \left(x \right)
F_{78}\! \left(x \right) = F_{79}\! \left(x \right)+F_{88}\! \left(x \right)
F_{79}\! \left(x \right) = F_{34}\! \left(x \right)+F_{35}\! \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_{39}\! \left(x \right)+F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{79}\! \left(x \right)+F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{4}\! \left(x \right) F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{86}\! \left(x \right)+F_{87}\! \left(x \right)
F_{86}\! \left(x \right) = F_{40}\! \left(x \right)
F_{87}\! \left(x \right) = F_{83}\! \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_{91}\! \left(x \right)+F_{92}\! \left(x \right)
F_{91}\! \left(x \right) = F_{74}\! \left(x \right)
F_{92}\! \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_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_6(x))
Eq(F_10(x), F_11(x) + F_23(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_17(x))
Eq(F_14(x), F_1(x) + F_15(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_14(x)*F_4(x))
Eq(F_17(x), F_11(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_22(x))
Eq(F_21(x), F_15(x))
Eq(F_22(x), F_18(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_7(x))
Eq(F_27(x), F_23(x))
Eq(F_28(x), F_29(x) + F_44(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_14(x) + F_32(x))
Eq(F_32(x), F_29(x) + F_33(x))
Eq(F_33(x), F_34(x) + F_35(x) + F_38(x))
Eq(F_34(x), 0)
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_21(x) + F_37(x))
Eq(F_37(x), F_33(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_29(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_21(x) + F_43(x))
Eq(F_43(x), F_40(x))
Eq(F_44(x), F_34(x) + F_45(x) + F_71(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_57(x))
Eq(F_47(x), F_48(x) + F_7(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_52(x))
Eq(F_51(x), F_15(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_56(x))
Eq(F_55(x), F_15(x))
Eq(F_56(x), F_53(x))
Eq(F_57(x), F_44(x) + F_58(x))
Eq(F_58(x), 2*F_34(x) + F_59(x) + F_63(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_61(x) + F_62(x))
Eq(F_61(x), F_48(x))
Eq(F_62(x), F_58(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_65(x) + F_66(x))
Eq(F_65(x), F_33(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), F_4(x)*F_68(x))
Eq(F_68(x), F_69(x) + F_70(x))
Eq(F_69(x), F_33(x))
Eq(F_70(x), F_67(x))
Eq(F_71(x), F_4(x)*F_72(x))
Eq(F_72(x), F_73(x) + F_78(x))
Eq(F_73(x), F_29(x) + F_74(x))
Eq(F_74(x), F_75(x))
Eq(F_75(x), F_4(x)*F_76(x))
Eq(F_76(x), F_26(x) + F_77(x))
Eq(F_77(x), F_74(x))
Eq(F_78(x), F_79(x) + F_88(x))
Eq(F_79(x), F_34(x) + F_35(x) + F_80(x))
Eq(F_80(x), F_4(x)*F_81(x))
Eq(F_81(x), F_39(x) + F_82(x))
Eq(F_82(x), F_79(x) + F_83(x))
Eq(F_83(x), F_84(x))
Eq(F_84(x), F_4(x)*F_85(x))
Eq(F_85(x), F_86(x) + F_87(x))
Eq(F_86(x), F_40(x))
Eq(F_87(x), F_83(x))
Eq(F_88(x), F_89(x))
Eq(F_89(x), F_4(x)*F_90(x))
Eq(F_90(x), F_91(x) + F_92(x))
Eq(F_91(x), F_74(x))
Eq(F_92(x), F_88(x))
Pack JSON:
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Specification JSON:
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