0123_0132_0213_0231_3102
Counting sequence:
1, 1, 2, 6, 19, 55, 144, 353, 832, 1914, 4337, 9735, 21722, 48285, 107066, 237018, 524123, 1158131, 2557728, 5646645, 12462704, 27501222, 60678161, 133865851, 295308450, 651417001, 1436896866, 3169420638, 6990773203, 15419276063, 34009319840, 75011592057, 165445985680, 364906995074, 804834810721, 1775130538591, 3915192231258, 8635258362661, 19045710511226, 42006715589266, 92648855122811, 204343688672699, 450694526430736, 994038609394733, 2192422042367136, 4835540447478798, 10665122475569377, 23522671801034931, 51880891828292866, 114426918718426385, 252376529602901122, 556633983985377654, 1227694940005475411, 2707766495881425863, 5972167115332094576, 13172029396521101137, 29051825654358927264, 64075819015336682058, 141323668383916494481, 311699163970200675447, 687474149460468818330, 1516271971357593672557, 3344243113242858344058, 7375960386556395367946, 16268192761638064589915, 35880628664296877563203, 79137217760095720711360, 174542628354552966265061, 384965885491071840558736, 849068988932631892542518, 1872680606527878272524913, 4130327099045282397492651, 9109723187829712220582178, 20092126983492272258623033, 44314581068141311992722018, 97738885327528790828938318, 215569897644077793617389971, 475454376365241293551299119, 1048647638072483711956218624, 2312865173812461945878300905, 5101184724028054247681050992, 11251017086189897998039938514, 24814899346291452795052939905, 54730983416771460481603304527, 120712983919992514458157676954, 266240867186696677852095790453, 587212717790844707823433504826, 1295138419502782017883390795842, 2856517706194040693034882103803, 6300248130181806161087568536203, 13895634679871054386668903406384, 30647787065943689580177435271005, 67595822262081385481857560963136, 149087279204053565624603893564190, 328822345474082761264020212507841, 725240513210298588718752844310627, 1599568305624734364205599430617346, 3527958956723686791527563780497697, 7781158426657891094769714960484354, 17161885158940870778593208613510630, 37851729274606001496557994824612755
Generating function in Maple syntax:
-(x^8+x^7-4*x^6-x^5+x^4-5*x^3+9*x^2-5*x+1)/(x^2+x-1)/(x^3+2*x-1)/(-1+x)^3
Generating function in latex syntax:
-\frac{x^{8}+x^{7}-4 x^{6}-x^{5}+x^{4}-5 x^{3}+9 x^{2}-5 x +1}{\left(x^{2}+x -1\right) \left(x^{3}+2 x -1\right) \left(-1+x \right)^{3}}
Generating function in sympy syntax:
(-x**8 - x**7 + 4*x**6 + x**5 - x**4 + 5*x**3 - 9*x**2 + 5*x - 1)/((x - 1)**3*(x**2 + x - 1)*(x**3 + 2*x - 1))
Implicit equation for the generating function in Maple syntax:
(x^2+x-1)*(x^3+2*x-1)*(-1+x)^3*F(x)+x^8+x^7-4*x^6-x^5+x^4-5*x^3+9*x^2-5*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}+x -1\right) \left(x^{3}+2 x -1\right) \left(-1+x \right)^{3} F \! \left(x \right)+x^{8}+x^{7}-4 x^{6}-x^{5}+x^{4}-5 x^{3}+9 x^{2}-5 x +1 = 0
Explicit closed form in Maple syntax:
7/4-1/2*n^2-n-1997/1180*Sum(_alpha^(-n+3),_alpha = RootOf(_Z^5+_Z^4+_Z^3+_Z^2-3*_Z+1))-1563/590*Sum(_alpha^(-n+2),_alpha = RootOf(_Z^5+_Z^4+_Z^3+_Z^2-3*_Z+1))-3899/1180*Sum(_alpha^(-n+1),_alpha = RootOf(_Z^5+_Z^4+_Z^3+_Z^2-3*_Z+1))-411/118*Sum(_alpha^(-n),_alpha = RootOf(_Z^5+_Z^4+_Z^3+_Z^2-3*_Z+1))+3941/1180*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^5+_Z^4+_Z^3+_Z^2-3*_Z+1))-piecewise(n = 0,1,0)
Explicit closed form in latex syntax:
\frac{7}{4}-\frac{n^{2}}{2}-n -\frac{1997 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}+\textit{\_Z}^{4}+\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +3}\right)}{1180}-\frac{1563 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}+\textit{\_Z}^{4}+\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{590}-\frac{3899 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}+\textit{\_Z}^{4}+\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{1180}-\frac{411 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}+\textit{\_Z}^{4}+\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{118}+\frac{3941 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}+\textit{\_Z}^{4}+\textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{1180}-\left(\left\{\begin{array}{cc}1 & n =0 \\ 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) = 55
a(6) = 144
a(7) = 353
a(8) = 832
a(n+5) = -n^2-a(n)-a(n+1)-a(n+2)-a(n+3)+3*a(n+4)-n+9, 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) = 55
a \! \left(6\right) = 144
a \! \left(7\right) = 353
a \! \left(8\right) = 832
a \! \left(n +5\right) = -n^{2}-a \! \left(n \right)-a \! \left(n +1\right)-a \! \left(n +2\right)-a \! \left(n +3\right)+3 a \! \left(n +4\right)-n +9, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15954/
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[12,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[7,x]
F[12,x] = F[13,x]+F[24,x]
F[13,x] = F[14,x]
F[14,x] = F[15,x]*F[4,x]
F[15,x] = F[16,x]+F[6,x]
F[16,x] = F[13,x]+F[17,x]
F[17,x] = F[18,x]
F[18,x] = F[19,x]*F[4,x]
F[19,x] = F[20,x]+F[23,x]
F[20,x] = F[13,x]+F[21,x]
F[21,x] = F[22,x]
F[22,x] = F[13,x]*F[4,x]
F[23,x] = F[17,x]
F[24,x] = F[25,x]+F[26,x]+F[58,x]
F[25,x] = 0
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[28,x]+F[35,x]
F[28,x] = F[29,x]+F[7,x]
F[29,x] = F[25,x]+F[30,x]+F[32,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[4,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[11,x]+F[34,x]
F[34,x] = F[29,x]
F[35,x] = F[24,x]+F[36,x]
F[36,x] = 2*F[25,x]+F[37,x]+F[48,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[39,x]
F[39,x] = F[22,x]+F[25,x]+F[40,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]+F[45,x]
F[42,x] = F[4,x]+F[43,x]
F[43,x] = F[25,x]+F[30,x]+F[44,x]
F[44,x] = F[4,x]*F[7,x]
F[45,x] = F[39,x]+F[46,x]
F[46,x] = 2*F[25,x]+F[37,x]+F[47,x]
F[47,x] = F[17,x]*F[4,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[50,x]+F[51,x]
F[50,x] = F[17,x]
F[51,x] = F[52,x]
F[52,x] = 2*F[25,x]+F[53,x]+F[55,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[21,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[23,x]+F[57,x]
F[57,x] = F[52,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[60,x]+F[61,x]
F[60,x] = F[13,x]+F[39,x]
F[61,x] = F[62,x]
F[62,x] = F[25,x]+F[63,x]+F[70,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[28,x]+F[65,x]
F[65,x] = F[17,x]+F[66,x]
F[66,x] = 2*F[25,x]+F[53,x]+F[67,x]
F[67,x] = F[4,x]*F[68,x]
F[68,x] = F[23,x]+F[69,x]
F[69,x] = F[66,x]
F[70,x] = F[4,x]*F[71,x]
F[71,x] = F[20,x]+F[72,x]
F[72,x] = F[62,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_{12}\! \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_{7}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{24}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{6}\! \left(x \right)
F_{16}\! \left(x \right) = F_{13}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{23}\! \left(x \right)
F_{20}\! \left(x \right) = F_{13}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{17}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{26}\! \left(x \right)+F_{58}\! \left(x \right)
F_{25}\! \left(x \right) = 0
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{35}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{7}\! \left(x \right)
F_{29}\! \left(x \right) = F_{25}\! \left(x \right)+F_{30}\! \left(x \right)+F_{32}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{11}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{29}\! \left(x \right)
F_{35}\! \left(x \right) = F_{24}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = 2 F_{25}\! \left(x \right)+F_{37}\! \left(x \right)+F_{48}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{22}\! \left(x \right)+F_{25}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{45}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{25}\! \left(x \right)+F_{30}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{45}\! \left(x \right) = F_{39}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = 2 F_{25}\! \left(x \right)+F_{37}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \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_{51}\! \left(x \right)
F_{50}\! \left(x \right) = F_{17}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = 2 F_{25}\! \left(x \right)+F_{53}\! \left(x \right)+F_{55}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{21}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{23}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{52}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{61}\! \left(x \right)
F_{60}\! \left(x \right) = F_{13}\! \left(x \right)+F_{39}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{25}\! \left(x \right)+F_{63}\! \left(x \right)+F_{70}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{28}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{17}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = 2 F_{25}\! \left(x \right)+F_{53}\! \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_{23}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{66}\! \left(x \right)
F_{70}\! \left(x \right) = F_{4}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{20}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{62}\! \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_12(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_7(x))
Eq(F_12(x), F_13(x) + F_24(x))
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_15(x)*F_4(x))
Eq(F_15(x), F_16(x) + F_6(x))
Eq(F_16(x), F_13(x) + F_17(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_19(x)*F_4(x))
Eq(F_19(x), F_20(x) + F_23(x))
Eq(F_20(x), F_13(x) + F_21(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_13(x)*F_4(x))
Eq(F_23(x), F_17(x))
Eq(F_24(x), F_25(x) + F_26(x) + F_58(x))
Eq(F_25(x), 0)
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_28(x) + F_35(x))
Eq(F_28(x), F_29(x) + F_7(x))
Eq(F_29(x), F_25(x) + F_30(x) + F_32(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_4(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_11(x) + F_34(x))
Eq(F_34(x), F_29(x))
Eq(F_35(x), F_24(x) + F_36(x))
Eq(F_36(x), 2*F_25(x) + F_37(x) + F_48(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_22(x) + F_25(x) + F_40(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_45(x))
Eq(F_42(x), F_4(x) + F_43(x))
Eq(F_43(x), F_25(x) + F_30(x) + F_44(x))
Eq(F_44(x), F_4(x)*F_7(x))
Eq(F_45(x), F_39(x) + F_46(x))
Eq(F_46(x), 2*F_25(x) + F_37(x) + F_47(x))
Eq(F_47(x), F_17(x)*F_4(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_51(x))
Eq(F_50(x), F_17(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), 2*F_25(x) + F_53(x) + F_55(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_21(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_23(x) + F_57(x))
Eq(F_57(x), F_52(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_60(x) + F_61(x))
Eq(F_60(x), F_13(x) + F_39(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_25(x) + F_63(x) + F_70(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_28(x) + F_65(x))
Eq(F_65(x), F_17(x) + F_66(x))
Eq(F_66(x), 2*F_25(x) + F_53(x) + F_67(x))
Eq(F_67(x), F_4(x)*F_68(x))
Eq(F_68(x), F_23(x) + F_69(x))
Eq(F_69(x), F_66(x))
Eq(F_70(x), F_4(x)*F_71(x))
Eq(F_71(x), F_20(x) + F_72(x))
Eq(F_72(x), F_62(x))
Pack JSON:
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Specification JSON:
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