0123_0132_0321_1032
Counting sequence:
1, 1, 2, 6, 20, 66, 220, 734, 2448, 8166, 27240, 90866, 303108, 1011098, 3372788, 11250838, 37530184, 125191982, 417611392, 1393054666, 4646907004, 15501003106, 51707748204, 172485045390, 575370073472, 1919301007798, 6402342645848, 21356728928034, 71241090290804, 237643740430698, 792724355219236, 2644344438523142, 8820919230637752, 29424539004794238, 98153431973105136, 327417065277701114, 1092187328349727212, 3643283404290399410, 12153147742552416284, 40540079829741973246, 135232295979369281008, 451103548701821433734, 1504776726429470722888, 5019585864310118936914, 16744173276102910405348, 55854675321644678008378, 186318231652488121921620, 621514371826591716304758, 2073227675901702128033576, 6915806280540896866310350, 23069524426046008199426144, 76954578490927440160012842, 256702610827565821289707868, 856300322838589118495941890, 2856419108982168600538694092, 9528351103630805861081706350, 31784367521058043659700339296, 106025272130109927203729534550, 353675696797054293930893261112, 1179780027835069240933946244482, 3935472317390255856617060850516, 13127822132542671250308562988042, 43791367349259230260614909536900, 146077836442040479406902459960294, 487281754173142828018396730704792, 1625458822045642851222090435635486, 5422153322053593400570340678497744, 18087044869495484922137704281912922, 60334183244239405362168750498462412, 201260830280176008706770115584526930, 671359412310155729603476120176960636, 2239499160715891633035091818050045406, 7470449358249527826691184188914698960, 24919684987214278825298580273802049062, 83126284655987944476415064588016970280, 277289989991995856815164774590667568050, 924975040902690738093825600255992325572, 3085502027381627457922459479399057416922, 10292518543727593079723538683867798203828, 34333452719483105721556571582077109217878, 114528428647753156226128061708986198623240, 382040253151706912563819376434105388457582, 1274397603734901813479695685225643876806208, 4251094587565199297785921350655868999063690, 14180664762286700306949022990209383865208380, 47303405971856788206985047253641453694579874, 157793182057952460598730694731338326194399276, 526361427732870471714554311736554078144858702, 1755819541703847263335480631214223246593138496, 5857006422958651145905756805603279236729513718, 19537613874196766251569843970297938382537739800, 65172944731787056220521660724238209474646154210, 217401815409108891532369621686187399550019439156, 725201991987393194777388856141891788033783184042, 2419105508354681999581918162895505096360939665892, 8069574442996973972614469645066118057717329312454, 26918227198514779329137976710863376258446671908984, 89792957562921207570055540228267968353350718337214, 299528463313566012012755230714942389183494276901296, 999157425816140074992230269793826276046845562325178, 3332957243927877885901225626902962433160488338396076
Generating function in Maple syntax:
(x^4-2*x^3-2*x^2-2*x+1)/(2*x^4-x^3-x^2-3*x+1)
Generating function in latex syntax:
\frac{x^{4}-2 x^{3}-2 x^{2}-2 x +1}{2 x^{4}-x^{3}-x^{2}-3 x +1}
Generating function in sympy syntax:
(x**4 - 2*x**3 - 2*x**2 - 2*x + 1)/(2*x**4 - x**3 - x**2 - 3*x + 1)
Implicit equation for the generating function in Maple syntax:
(2*x^4-x^3-x^2-3*x+1)*F(x)-x^4+2*x^3+2*x^2+2*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x^{4}-x^{3}-x^{2}-3 x +1\right) F \! \left(x \right)-x^{4}+2 x^{3}+2 x^{2}+2 x -1 = 0
Explicit closed form in Maple syntax:
368/4155*Sum(_alpha^(-n+2),_alpha = RootOf(2*_Z^4-_Z^3-_Z^2-3*_Z+1))+219/1385*Sum(_alpha^(-n+1),_alpha = RootOf(2*_Z^4-_Z^3-_Z^2-3*_Z+1))+58/831*Sum(_alpha^(-n),_alpha = RootOf(2*_Z^4-_Z^3-_Z^2-3*_Z+1))+43/4155*Sum(_alpha^(-n-1),_alpha = RootOf(2*_Z^4-_Z^3-_Z^2-3*_Z+1))+piecewise(n = 0,1/2,0)
Explicit closed form in latex syntax:
\frac{368 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \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)}{4155}+\frac{219 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \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)}{1385}+\frac{58 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \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)}{831}+\frac{43 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \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)}{4155}+\left(\left\{\begin{array}{cc}\frac{1}{2} & 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) = 20
a(n+4) = -2*a(n)+a(n+1)+a(n+2)+3*a(n+3), 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 +4\right) = -2 a \! \left(n \right)+a \! \left(n +1\right)+a \! \left(n +2\right)+3 a \! \left(n +3\right), \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14807/
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[19,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[14,x]
F[12,x] = F[13,x]
F[13,x] = F[10,x]*F[4,x]
F[14,x] = F[15,x]+F[16,x]+F[18,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[4,x]
F[18,x] = F[12,x]*F[4,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[15,x]+F[21,x]+F[69,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[29,x]
F[23,x] = F[12,x]+F[24,x]
F[24,x] = F[15,x]+F[25,x]+F[28,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[17,x]+F[27,x]
F[27,x] = F[14,x]
F[28,x] = F[11,x]*F[4,x]
F[29,x] = F[30,x]+F[58,x]
F[30,x] = F[15,x]+F[21,x]+F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[2,x]+F[33,x]
F[33,x] = F[15,x]+F[34,x]+F[57,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[36,x]+F[38,x]
F[36,x] = F[37,x]+F[4,x]
F[37,x] = F[15,x]+F[18,x]+F[25,x]
F[38,x] = F[33,x]+F[39,x]
F[39,x] = F[15,x]+F[40,x]+F[54,x]+F[55,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]+F[46,x]
F[42,x] = F[43,x]+F[44,x]
F[43,x] = F[16,x]
F[44,x] = F[45,x]
F[45,x] = F[27,x]*F[4,x]
F[46,x] = F[47,x]+F[50,x]
F[47,x] = 2*F[15,x]+F[40,x]+F[48,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[33,x]
F[50,x] = F[51,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[53,x]
F[53,x] = F[15,x]+F[40,x]+F[48,x]+F[54,x]
F[54,x] = F[30,x]*F[4,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[49,x]+F[52,x]
F[57,x] = F[2,x]*F[4,x]
F[58,x] = F[15,x]+F[55,x]+F[59,x]+F[67,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[61,x]+F[64,x]
F[61,x] = F[62,x]+F[63,x]
F[62,x] = F[16,x]
F[63,x] = F[45,x]
F[64,x] = F[65,x]+F[66,x]
F[65,x] = 2*F[15,x]+F[48,x]+F[59,x]
F[66,x] = F[51,x]
F[67,x] = F[4,x]*F[68,x]
F[68,x] = F[30,x]+F[53,x]
F[69,x] = F[4,x]*F[70,x]
F[70,x] = F[32,x]+F[68,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_{19}\! \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_{14}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \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_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{15}\! \left(x \right)+F_{21}\! \left(x \right)+F_{69}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{29}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{15}\! \left(x \right)+F_{25}\! \left(x \right)+F_{28}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{17}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{14}\! \left(x \right)
F_{28}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{58}\! \left(x \right)
F_{30}\! \left(x \right) = F_{15}\! \left(x \right)+F_{21}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{2}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{15}\! \left(x \right)+F_{34}\! \left(x \right)+F_{57}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{38}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{15}\! \left(x \right)+F_{18}\! \left(x \right)+F_{25}\! \left(x \right)
F_{38}\! \left(x \right) = F_{33}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{15}\! \left(x \right)+F_{40}\! \left(x \right)+F_{54}\! \left(x \right)+F_{55}\! \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_{46}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{44}\! \left(x \right)
F_{43}\! \left(x \right) = F_{16}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{50}\! \left(x \right)
F_{47}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{40}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{33}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{15}\! \left(x \right)+F_{40}\! \left(x \right)+F_{48}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{49}\! \left(x \right)+F_{52}\! \left(x \right)
F_{57}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{58}\! \left(x \right) = F_{15}\! \left(x \right)+F_{55}\! \left(x \right)+F_{59}\! \left(x \right)+F_{67}\! \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_{64}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{63}\! \left(x \right)
F_{62}\! \left(x \right) = F_{16}\! \left(x \right)
F_{63}\! \left(x \right) = F_{45}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{66}\! \left(x \right)
F_{65}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{48}\! \left(x \right)+F_{59}\! \left(x \right)
F_{66}\! \left(x \right) = F_{51}\! \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_{53}\! \left(x \right)
F_{69}\! \left(x \right) = F_{4}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{32}\! \left(x \right)+F_{68}\! \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_19(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_14(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_10(x)*F_4(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_18(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_4(x))
Eq(F_18(x), F_12(x)*F_4(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_15(x) + F_21(x) + F_69(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_29(x))
Eq(F_23(x), F_12(x) + F_24(x))
Eq(F_24(x), F_15(x) + F_25(x) + F_28(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_17(x) + F_27(x))
Eq(F_27(x), F_14(x))
Eq(F_28(x), F_11(x)*F_4(x))
Eq(F_29(x), F_30(x) + F_58(x))
Eq(F_30(x), F_15(x) + F_21(x) + F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_2(x) + F_33(x))
Eq(F_33(x), F_15(x) + F_34(x) + F_57(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_36(x) + F_38(x))
Eq(F_36(x), F_37(x) + F_4(x))
Eq(F_37(x), F_15(x) + F_18(x) + F_25(x))
Eq(F_38(x), F_33(x) + F_39(x))
Eq(F_39(x), F_15(x) + F_40(x) + F_54(x) + F_55(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_46(x))
Eq(F_42(x), F_43(x) + F_44(x))
Eq(F_43(x), F_16(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_27(x)*F_4(x))
Eq(F_46(x), F_47(x) + F_50(x))
Eq(F_47(x), 2*F_15(x) + F_40(x) + F_48(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_33(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_15(x) + F_40(x) + F_48(x) + F_54(x))
Eq(F_54(x), F_30(x)*F_4(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_49(x) + F_52(x))
Eq(F_57(x), F_2(x)*F_4(x))
Eq(F_58(x), F_15(x) + F_55(x) + F_59(x) + F_67(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_61(x) + F_64(x))
Eq(F_61(x), F_62(x) + F_63(x))
Eq(F_62(x), F_16(x))
Eq(F_63(x), F_45(x))
Eq(F_64(x), F_65(x) + F_66(x))
Eq(F_65(x), 2*F_15(x) + F_48(x) + F_59(x))
Eq(F_66(x), F_51(x))
Eq(F_67(x), F_4(x)*F_68(x))
Eq(F_68(x), F_30(x) + F_53(x))
Eq(F_69(x), F_4(x)*F_70(x))
Eq(F_70(x), F_32(x) + F_68(x))
Pack JSON:
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Specification JSON:
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