0213_1032_1230_1302
Counting sequence:
1, 1, 2, 6, 20, 63, 189, 550, 1568, 4403, 12220, 33603, 91718, 248825, 671667, 1805487, 4836181, 12915505, 34404218, 91445290, 242600756, 642561892, 1699509595, 4489500194, 11846946454, 31232563642, 82272040261, 216562951027, 569693627374, 1497810882774, 3936043640763, 10338924810156, 27147228813677, 71257350781324, 186984199704339, 490530696576899, 1286550071564337, 3373634542413681, 8844849713738661, 23185315205931008, 60767820433614140, 159250014627395286, 417288780594032492, 1093328986055676541, 2864354908517482510, 7503587171673739198, 19655360100046643304, 51483307475685464335, 134842949813758722333, 353158029224941981538, 924892639012842387260, 2422127804236957090063, 6342904007923945561308, 16609869590517660974335, 43494342324548920056182, 113890912558149101195097, 298219671124101405523271, 780864055347644060049915, 2044595562145911566664633, 5353441199478558148702225, 14016933561534492766315646, 36700161991309255585387202, 96090067448845288030284460, 251585185977458709643370304, 658700699783507451667804319, 1724598765064981289428606502, 4515285877509468228425655126, 11821701219673660153902128862, 30950846125638076527895325389, 81033227767297820609327729059, 212154394670384247007544089178, 555442875850253567471200051066, 1454204267321371924344374028767, 3807239747718180179968541131684, 9967677291370530247962366238729, 26096169464237854981246987973684, 68321708305405962657368274201427, 178870994704017289673371120281991, 468296016492476264904434020949701, 1226028075530890995261560850698893, 3209813830253011731144273672583405, 8403472974857460607061764709402196, 22000743553659168786406113707934732, 57599079565033624033734872228071326, 150797243419132157175693708455015000, 394794390226946851209240371198153833, 1033589971188993078159497644179184794, 2705984924330408874820770247814990870, 7084386656453377661258319798674262348, 18547225850929688152425144112521777423, 48557409005723667188070481719491654925, 127125275737256470615960005358535454398, 332819056506215219531694863896974320648, 871333377649255278578486535510402395595, 2281184526015825095462253635399140572572, 5972228219685480739256902480166572277707, 15635518775621716448736722297187434602486, 40934371445922719602599302561462108350825, 107167696312500657424591390962703857184923, 280568951707736895202901679245118468108535, 734539703297217576643879700527115005205661
Generating function in Maple syntax:
(x^5-3*x^4+4*x^3-8*x^2+5*x-1)/(x^2-3*x+1)/(x^3-2*x^2+3*x-1)
Generating function in latex syntax:
\frac{x^{5}-3 x^{4}+4 x^{3}-8 x^{2}+5 x -1}{\left(x^{2}-3 x +1\right) \left(x^{3}-2 x^{2}+3 x -1\right)}
Generating function in sympy syntax:
(x**5 - 3*x**4 + 4*x**3 - 8*x**2 + 5*x - 1)/((x**2 - 3*x + 1)*(x**3 - 2*x**2 + 3*x - 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(x^3-2*x^2+3*x-1)*F(x)-x^5+3*x^4-4*x^3+8*x^2-5*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(x^{3}-2 x^{2}+3 x -1\right) F \! \left(x \right)-x^{5}+3 x^{4}-4 x^{3}+8 x^{2}-5 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/13800*(391*2^(1/3)*((I-43/391*23^(1/2))*3^(1/2)-129/391*I*23^(1/2)+1)*(11+3*23^(1/2)*3^(1/2))^(2/3)-4600-460*2^(2/3)*((I-7/46*23^(1/2))*3^(1/2)+21/46*I*23^(1/2)-1)*(11+3*23^(1/2)*3^(1/2))^(1/3))*(11/600*2^(1/3)*((I-3/11*23^(1/2))*3^(1/2)-9/11*I*23^(1/2)+1)*(11+3*23^(1/2)*3^(1/2))^(2/3)-1/12*I*3^(1/2)*(44+12*23^(1/2)*3^(1/2))^(1/3)+1/12*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+1/13800*(460*2^(2/3)*((I+7/46*23^(1/2))*3^(1/2)+21/46*I*23^(1/2)+1)*(11+3*23^(1/2)*3^(1/2))^(1/3)-4600-391*((I+43/391*23^(1/2))*3^(1/2)-129/391*I*23^(1/2)-1)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3))*(-11/600*2^(1/3)*((I+3/11*23^(1/2))*3^(1/2)-9/11*I*23^(1/2)-1)*(11+3*23^(1/2)*3^(1/2))^(2/3)+1/12*I*3^(1/2)*(44+12*23^(1/2)*3^(1/2))^(1/3)+1/12*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+1/13800*((-140*3^(1/2)*2^(2/3)*23^(1/2)-920*2^(2/3))*(11+3*23^(1/2)*3^(1/2))^(1/3)-4600+(86*2^(1/3)*3^(1/2)*23^(1/2)-782*2^(1/3))*(11+3*23^(1/2)*3^(1/2))^(2/3))*(1/300*2^(1/3)*(3*23^(1/2)*3^(1/2)-11)*(11+3*23^(1/2)*3^(1/2))^(2/3)-1/6*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+1/13800*(4140*5^(1/2)+6900)*(3/2-1/2*5^(1/2))^(-n)+1/13800*(-4140*5^(1/2)+6900)*(3/2+1/2*5^(1/2))^(-n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(391 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}-\frac{43 \sqrt{23}}{391}\right) \sqrt{3}-\frac{129 \,\mathrm{I} \sqrt{23}}{391}+1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}-4600-460 \,2^{\frac{2}{3}} \left(\left(\mathrm{I}-\frac{7 \sqrt{23}}{46}\right) \sqrt{3}+\frac{21 \,\mathrm{I} \sqrt{23}}{46}-1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{11 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}-\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}+1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}-\frac{\mathrm{I} \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{13800}\\+\\\frac{\left(460 \,2^{\frac{2}{3}} \left(\left(\mathrm{I}+\frac{7 \sqrt{23}}{46}\right) \sqrt{3}+\frac{21 \,\mathrm{I} \sqrt{23}}{46}+1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}-4600-391 \left(\left(\mathrm{I}+\frac{43 \sqrt{23}}{391}\right) \sqrt{3}-\frac{129 \,\mathrm{I} \sqrt{23}}{391}-1\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(-\frac{11 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}+\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}-1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}+\frac{\mathrm{I} \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{13800}\\+\\\frac{\left(\left(-140 \sqrt{3}\, 2^{\frac{2}{3}} \sqrt{23}-920 \,2^{\frac{2}{3}}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}-4600+\left(86 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{23}-782 \,2^{\frac{1}{3}}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{2^{\frac{1}{3}} \left(3 \sqrt{23}\, \sqrt{3}-11\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{300}-\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{2}{3}\right)^{-n}}{13800}\\+\frac{\left(4140 \sqrt{5}+6900\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{13800}+\frac{\left(-4140 \sqrt{5}+6900\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{13800} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 63
a(n+5) = a(n)-5*a(n+1)+10*a(n+2)-12*a(n+3)+6*a(n+4), n >= 6
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(5\right) = 63
a \! \left(n +5\right) = a \! \left(n \right)-5 a \! \left(n +1\right)+10 a \! \left(n +2\right)-12 a \! \left(n +3\right)+6 a \! \left(n +4\right), \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21373/
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[16,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[16,x]*F[7,x]
F[7,x] = F[28,x]+F[8,x]
F[8,x] = F[20,x]+F[9,x]
F[9,x] = F[10,x]*F[17,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]
F[13,x] = F[10,x]*F[14,x]*F[16,x]
F[14,x] = F[1,x]+F[15,x]
F[15,x] = F[14,x]*F[16,x]
F[16,x] = x
F[17,x] = F[1,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[10,x]*F[16,x]*F[17,x]
F[20,x] = F[14,x]*F[21,x]*F[23,x]
F[21,x] = F[1,x]+F[22,x]
F[22,x] = F[16,x]*F[21,x]
F[23,x] = F[24,x]
F[24,x] = F[16,x]*F[25,x]
F[25,x] = F[21,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[14,x]*F[16,x]*F[21,x]*F[25,x]
F[28,x] = F[14,x]^2*F[29,x]
F[29,x] = F[30,x]+F[31,x]
F[30,x] = F[21,x]*F[23,x]
F[31,x] = F[32,x]
F[32,x] = F[16,x]*F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[16,x]*F[35,x]*F[49,x]
F[35,x] = F[1,x]+F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[16,x]*F[38,x]
F[38,x] = F[14,x]+F[39,x]
F[39,x] = F[36,x]+F[40,x]
F[40,x] = F[41,x]+F[42,x]+F[48,x]
F[41,x] = 0
F[42,x] = F[16,x]*F[43,x]
F[43,x] = F[44,x]+F[45,x]
F[44,x] = F[16,x]
F[45,x] = F[46,x]
F[46,x] = F[41,x]+F[42,x]+F[47,x]
F[47,x] = F[16,x]*F[36,x]
F[48,x] = F[16,x]*F[39,x]
F[49,x] = F[50,x]+F[54,x]
F[50,x] = F[51,x]+F[52,x]
F[51,x] = F[12,x]*F[17,x]*F[21,x]
F[52,x] = F[21,x]*F[23,x]*F[53,x]
F[53,x] = F[15,x]
F[54,x] = F[14,x]*F[21,x]*F[23,x]*F[53,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_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{0}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{16}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{28}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{20}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right) F_{17}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{10}\! \left(x \right) F_{14}\! \left(x \right) F_{16}\! \left(x \right)
F_{14}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{14}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = x
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{10}\! \left(x \right) F_{16}\! \left(x \right) F_{17}\! \left(x \right)
F_{20}\! \left(x \right) = F_{14}\! \left(x \right) F_{21}\! \left(x \right) F_{23}\! \left(x \right)
F_{21}\! \left(x \right) = F_{1}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{16}\! \left(x \right) F_{21}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{16}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{21}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{14}\! \left(x \right) F_{16}\! \left(x \right) F_{21}\! \left(x \right) F_{25}\! \left(x \right)
F_{28}\! \left(x \right) = F_{14} \left(x \right)^{2} 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_{21}\! \left(x \right) F_{23}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{16}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{16}\! \left(x \right) F_{35}\! \left(x \right) F_{49}\! \left(x \right)
F_{35}\! \left(x \right) = F_{1}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{16}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{14}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{36}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{42}\! \left(x \right)+F_{48}\! \left(x \right)
F_{41}\! \left(x \right) = 0
F_{42}\! \left(x \right) = F_{16}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{45}\! \left(x \right)
F_{44}\! \left(x \right) = F_{16}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{41}\! \left(x \right)+F_{42}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{16}\! \left(x \right) F_{36}\! \left(x \right)
F_{48}\! \left(x \right) = F_{16}\! \left(x \right) F_{39}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{54}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{52}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right) F_{21}\! \left(x \right)
F_{52}\! \left(x \right) = F_{21}\! \left(x \right) F_{23}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{15}\! \left(x \right)
F_{54}\! \left(x \right) = F_{14}\! \left(x \right) F_{21}\! \left(x \right) F_{23}\! \left(x \right) F_{53}\! \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_16(x)*F_4(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_16(x)*F_7(x))
Eq(F_7(x), F_28(x) + F_8(x))
Eq(F_8(x), F_20(x) + F_9(x))
Eq(F_9(x), F_10(x)*F_17(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_10(x)*F_14(x)*F_16(x))
Eq(F_14(x), F_1(x) + F_15(x))
Eq(F_15(x), F_14(x)*F_16(x))
Eq(F_16(x), x)
Eq(F_17(x), F_1(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_10(x)*F_16(x)*F_17(x))
Eq(F_20(x), F_14(x)*F_21(x)*F_23(x))
Eq(F_21(x), F_1(x) + F_22(x))
Eq(F_22(x), F_16(x)*F_21(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_16(x)*F_25(x))
Eq(F_25(x), F_21(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_14(x)*F_16(x)*F_21(x)*F_25(x))
Eq(F_28(x), F_14(x)**2*F_29(x))
Eq(F_29(x), F_30(x) + F_31(x))
Eq(F_30(x), F_21(x)*F_23(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_16(x)*F_33(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_16(x)*F_35(x)*F_49(x))
Eq(F_35(x), F_1(x) + F_36(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_16(x)*F_38(x))
Eq(F_38(x), F_14(x) + F_39(x))
Eq(F_39(x), F_36(x) + F_40(x))
Eq(F_40(x), F_41(x) + F_42(x) + F_48(x))
Eq(F_41(x), 0)
Eq(F_42(x), F_16(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_45(x))
Eq(F_44(x), F_16(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_41(x) + F_42(x) + F_47(x))
Eq(F_47(x), F_16(x)*F_36(x))
Eq(F_48(x), F_16(x)*F_39(x))
Eq(F_49(x), F_50(x) + F_54(x))
Eq(F_50(x), F_51(x) + F_52(x))
Eq(F_51(x), F_12(x)*F_17(x)*F_21(x))
Eq(F_52(x), F_21(x)*F_23(x)*F_53(x))
Eq(F_53(x), F_15(x))
Eq(F_54(x), F_14(x)*F_21(x)*F_23(x)*F_53(x))
Pack JSON:
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Specification JSON:
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