0123_0132_0231_2031_3102
Counting sequence:
1, 1, 2, 6, 19, 53, 135, 330, 788, 1852, 4305, 9927, 22749, 51872, 117790, 266546, 601375, 1353337, 3038787, 6810054, 15235624, 34034360, 75927389, 169187595, 376603897, 837523836, 1861012530, 4132183438, 9168999627, 20333229565, 45067078079, 99840026754, 221087024956, 489388611700, 1082912217513, 2395501394639, 5297571270229, 11712414496408, 25889049864262, 57213109952298, 126413512308023, 279265830294273, 616844282168699, 1362301099901054, 3008266076607568, 6642172528406128, 14664238342757813, 32371927134212115, 71456395541008113, 157717766913129460, 348088935937181786, 768197217368793094, 1695238101557558371, 3740816938865983877, 8254334694728131447, 18212914690268562362, 40184660717912590692, 88659684927572276908, 195604342139451044225, 431537460184890535127, 952022835673505059021, 2100226474238764585872, 4633143330167026553966, 10220615339016771861026, 22546068838290735695951, 49734504378785352721801, 109708070840661186856371, 241999104007760528512054, 533806499370601247952504, 1177468643534453359174440, 2597231538981846599686925, 5728859873144653152978203, 12636368981444477076434409, 27872330685112235575162732, 61478243609851993948517506, 135602300934114204263896958, 299095822019272122683811579, 659707666580259196477850381, 1455093191958358511543017007, 3209433321663440974416684018, 7078876541362044802604895116, 15613450737592255431340160484, 34437543722667566466271711321, 75956381838336406993497730271, 167531050117543527935034445893, 369509315364311539369738252680, 814994355380073319799769534614, 1797558446503917835668164112922, 3964703579624602546973247673959, 8744556257134189086280627273233, 19286980445782117353298143440619, 42539283441208479943706984117486, 93824361079590434353969494632608, 206938178485041556821786930954464, 456420592171448735108380442523493, 1006675448942802187612929572673571, 2220308883411274498132044462289377, 4897077973075254863541266139077604, 10800910623255826179033055394779306, 23822288586247955384873342339748854, 52541972058221222690638324994376851
Generating function in Maple syntax:
(2*x^7-x^6-6*x^5+4*x^4-6*x^3+9*x^2-5*x+1)/(2*x-1)/(x^3+2*x-1)/(x-1)^2
Generating function in latex syntax:
\frac{2 x^{7}-x^{6}-6 x^{5}+4 x^{4}-6 x^{3}+9 x^{2}-5 x +1}{\left(2 x -1\right) \left(x^{3}+2 x -1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(2*x**7 - x**6 - 6*x**5 + 4*x**4 - 6*x**3 + 9*x**2 - 5*x + 1)/((x - 1)**2*(2*x - 1)*(x**3 + 2*x - 1))
Implicit equation for the generating function in Maple syntax:
-(2*x-1)*(x^3+2*x-1)*(x-1)^2*F(x)+2*x^7-x^6-6*x^5+4*x^4-6*x^3+9*x^2-5*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right) \left(x^{3}+2 x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+2 x^{7}-x^{6}-6 x^{5}+4 x^{4}-6 x^{3}+9 x^{2}-5 x +1 = 0
Explicit closed form in Maple syntax:
-n-970/59*Sum(_alpha^(-n+2),_alpha = RootOf(2*_Z^4-_Z^3+4*_Z^2-4*_Z+1))+23/59*Sum(_alpha^(1-n),_alpha = RootOf(2*_Z^4-_Z^3+4*_Z^2-4*_Z+1))-1967/59*Sum(_alpha^(-n),_alpha = RootOf(2*_Z^4-_Z^3+4*_Z^2-4*_Z+1))+1040/59*Sum(_alpha^(-n-1),_alpha = RootOf(2*_Z^4-_Z^3+4*_Z^2-4*_Z+1))+piecewise(n = 0,2,n = 1,1,0)
Explicit closed form in latex syntax:
-n -\frac{970 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{59}+\frac{23 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{1-n}\right)}{59}-\frac{1967 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{59}+\frac{1040 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-\textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{59}+\left(\left\{\begin{array}{cc}2 & n =0 \\ 1 & n =1 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 53
a(6) = 135
a(7) = 330
a(n+4) = -2*a(n)+a(n+1)-4*a(n+2)+4*a(n+3)-2*n+1, 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) = 53
a \! \left(6\right) = 135
a \! \left(7\right) = 330
a \! \left(n +4\right) = -2 a \! \left(n \right)+a \! \left(n +1\right)-4 a \! \left(n +2\right)+4 a \! \left(n +3\right)-2 n +1, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16163/
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[21,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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[16,x]
F[12,x] = F[13,x]
F[13,x] = F[14,x]*F[4,x]
F[14,x] = F[10,x]+F[15,x]
F[15,x] = F[12,x]
F[16,x] = F[17,x]
F[17,x] = F[18,x]*F[4,x]
F[18,x] = F[19,x]+F[20,x]
F[19,x] = F[4,x]
F[20,x] = F[16,x]
F[21,x] = F[22,x]+F[34,x]
F[22,x] = F[23,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[26,x]
F[25,x] = F[1,x]+F[12,x]
F[26,x] = F[22,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[30,x]+F[33,x]
F[30,x] = F[22,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[22,x]*F[4,x]
F[33,x] = F[27,x]
F[34,x] = F[35,x]+F[36,x]+F[52,x]
F[35,x] = 0
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[40,x]
F[38,x] = F[39,x]+F[7,x]
F[39,x] = F[17,x]
F[40,x] = F[34,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[44,x]+F[51,x]
F[44,x] = F[45,x]
F[45,x] = F[32,x]+F[35,x]+F[46,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]+F[49,x]
F[48,x] = F[16,x]+F[4,x]
F[49,x] = F[45,x]+F[50,x]
F[50,x] = F[42,x]
F[51,x] = F[50,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[54,x]+F[55,x]
F[54,x] = F[22,x]+F[45,x]
F[55,x] = F[56,x]+F[70,x]
F[56,x] = F[35,x]+F[57,x]+F[67,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[59,x]+F[63,x]
F[59,x] = F[12,x]+F[60,x]
F[60,x] = F[61,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[19,x]
F[63,x] = F[27,x]+F[64,x]
F[64,x] = F[65,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[31,x]
F[67,x] = F[4,x]*F[68,x]
F[68,x] = F[30,x]+F[69,x]
F[69,x] = F[56,x]
F[70,x] = 2*F[35,x]+F[71,x]+F[75,x]
F[71,x] = F[4,x]*F[72,x]
F[72,x] = F[73,x]+F[74,x]
F[73,x] = F[16,x]
F[74,x] = F[50,x]
F[75,x] = F[4,x]*F[76,x]
F[76,x] = F[44,x]+F[77,x]
F[77,x] = F[70,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_{21}\! \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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{16}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{10}\! \left(x \right)+F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)
F_{19}\! \left(x \right) = F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{16}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{34}\! \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_{25}\! \left(x \right)+F_{26}\! \left(x \right)
F_{25}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{26}\! \left(x \right) = F_{22}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \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_{33}\! \left(x \right)
F_{30}\! \left(x \right) = F_{22}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{27}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{36}\! \left(x \right)+F_{52}\! \left(x \right)
F_{35}\! \left(x \right) = 0
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_{40}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{7}\! \left(x \right)
F_{39}\! \left(x \right) = F_{17}\! \left(x \right)
F_{40}\! \left(x \right) = F_{34}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{51}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{32}\! \left(x \right)+F_{35}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{49}\! \left(x \right)
F_{48}\! \left(x \right) = F_{16}\! \left(x \right)+F_{4}\! \left(x \right)
F_{49}\! \left(x \right) = F_{45}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{42}\! \left(x \right)
F_{51}\! \left(x \right) = F_{50}\! \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_{55}\! \left(x \right)
F_{54}\! \left(x \right) = F_{22}\! \left(x \right)+F_{45}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{70}\! \left(x \right)
F_{56}\! \left(x \right) = F_{35}\! \left(x \right)+F_{57}\! \left(x \right)+F_{67}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{63}\! \left(x \right)
F_{59}\! \left(x \right) = F_{12}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{19}\! \left(x \right)
F_{63}\! \left(x \right) = F_{27}\! \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_{31}\! \left(x \right)
F_{67}\! \left(x \right) = F_{4}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{30}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{56}\! \left(x \right)
F_{70}\! \left(x \right) = 2 F_{35}\! \left(x \right)+F_{71}\! \left(x \right)+F_{75}\! \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_{74}\! \left(x \right)
F_{73}\! \left(x \right) = F_{16}\! \left(x \right)
F_{74}\! \left(x \right) = F_{50}\! \left(x \right)
F_{75}\! \left(x \right) = F_{4}\! \left(x \right) F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{44}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{70}\! \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_21(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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_16(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_14(x)*F_4(x))
Eq(F_14(x), F_10(x) + F_15(x))
Eq(F_15(x), F_12(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_18(x)*F_4(x))
Eq(F_18(x), F_19(x) + F_20(x))
Eq(F_19(x), F_4(x))
Eq(F_20(x), F_16(x))
Eq(F_21(x), F_22(x) + F_34(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_26(x))
Eq(F_25(x), F_1(x) + F_12(x))
Eq(F_26(x), F_22(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_30(x) + F_33(x))
Eq(F_30(x), F_22(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_22(x)*F_4(x))
Eq(F_33(x), F_27(x))
Eq(F_34(x), F_35(x) + F_36(x) + F_52(x))
Eq(F_35(x), 0)
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_40(x))
Eq(F_38(x), F_39(x) + F_7(x))
Eq(F_39(x), F_17(x))
Eq(F_40(x), F_34(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_51(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_32(x) + F_35(x) + F_46(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_49(x))
Eq(F_48(x), F_16(x) + F_4(x))
Eq(F_49(x), F_45(x) + F_50(x))
Eq(F_50(x), F_42(x))
Eq(F_51(x), F_50(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_55(x))
Eq(F_54(x), F_22(x) + F_45(x))
Eq(F_55(x), F_56(x) + F_70(x))
Eq(F_56(x), F_35(x) + F_57(x) + F_67(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_59(x) + F_63(x))
Eq(F_59(x), F_12(x) + F_60(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_19(x))
Eq(F_63(x), F_27(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_31(x))
Eq(F_67(x), F_4(x)*F_68(x))
Eq(F_68(x), F_30(x) + F_69(x))
Eq(F_69(x), F_56(x))
Eq(F_70(x), 2*F_35(x) + F_71(x) + F_75(x))
Eq(F_71(x), F_4(x)*F_72(x))
Eq(F_72(x), F_73(x) + F_74(x))
Eq(F_73(x), F_16(x))
Eq(F_74(x), F_50(x))
Eq(F_75(x), F_4(x)*F_76(x))
Eq(F_76(x), F_44(x) + F_77(x))
Eq(F_77(x), F_70(x))
Pack JSON:
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Specification JSON:
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