0213_1230_1302_3012
Counting sequence:
1, 1, 2, 6, 20, 61, 171, 454, 1167, 2944, 7349, 18247, 45220, 112124, 278640, 694834, 1739958, 4377125, 11063173, 28090814, 71635585, 183407096, 471244527, 1214596511, 3138989030, 8131065636, 21103210906, 54859797674, 142805090872, 372146185869, 970680673439, 2533721815302, 6617599866515, 17292234234144, 45203268256105, 118201606411719, 309161032804488, 808783275788860, 2116158002403908, 5537560427636274, 14492125233985370, 37929744303381541, 99278415978477177, 259867020725067166, 680243481359512965, 1780700695632548536, 4661524354003275491, 12203186271121531519, 31946627084477750602, 83633809863800419844, 218948891532412567854, 573200761309438707082, 1500628622597952996060, 3928634440988612342189, 10285171117576454486995, 26926667242558264682054, 70494398264534111966743, 184555644845517106574720, 483170734832166259531293, 1264952884713685737665159, 3311680425319111008927980, 8670073115033711248363836, 22698507790901398351260952, 59425386846987730428799346, 155577523622853276968237950, 407306921155469050114764261, 1066342704887975735799016301, 2791720105150557808418695614, 7308815396954408844310830857, 19134721584707114739383154744, 50095340207581874813900981719, 131151280443720483403091709087, 343358463344647712438212381230, 898924032851767307279810654884, 2353413479372560274702927471810, 6161316088867359164562742380714, 16130534144987675548713550542848, 42230285042722518209555830200205, 110560318338654648672952620317703, 289450664608633103269385067921094, 757791664606912284603540011013915, 1983924307149207542574252526653856, 5193981212110455017377938104736977, 13598019238512721038462373684508551, 35600076319670983516585548393361360, 93202209348151077069988485685981436, 244006550970412536253851303647833836, 638817442035004295698674148428814834, 1672445772039750252628720454390697762, 4378519867817175013305249573066356069, 11463113818722889384611880447092237409, 30010821562664237325358751064053414942, 78569350817276340941478401199304672525, 205697230783939882159819129167434922104, 538522341321617618780893579020350601675, 1409869792750109840511549272228721523199, 3691087036057202115096846524136830380178, 9663391313658669888836608788725416583940, 25299086901353540238271084650329941463638, 66233869383192188037178590441251908178954, 173402521233645041970640048556213696059300
Generating function in Maple syntax:
(4*x^8-23*x^7+56*x^6-101*x^5+125*x^4-95*x^3+42*x^2-10*x+1)/(x^2-3*x+1)/(2*x-1)^2/(x-1)^4
Generating function in latex syntax:
\frac{4 x^{8}-23 x^{7}+56 x^{6}-101 x^{5}+125 x^{4}-95 x^{3}+42 x^{2}-10 x +1}{\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{4}}
Generating function in sympy syntax:
(4*x**8 - 23*x**7 + 56*x**6 - 101*x**5 + 125*x**4 - 95*x**3 + 42*x**2 - 10*x + 1)/((x - 1)**4*(2*x - 1)**2*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(2*x-1)^2*(x-1)^4*F(x)-4*x^8+23*x^7-56*x^6+101*x^5-125*x^4+95*x^3-42*x^2+10*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{4} F \! \left(x \right)-4 x^{8}+23 x^{7}-56 x^{6}+101 x^{5}-125 x^{4}+95 x^{3}-42 x^{2}+10 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/30*(-3*5^(1/2)+15)*(3/2-1/2*5^(1/2))^(-n)+1/30*(3*5^(1/2)+15)*(3/2+1/2*5^(1/2))^(-n)+1/6*n^3-n^2+2^(-1+n)*n+17/6*n-2^(n+1)+1)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(-3 \sqrt{5}+15\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{30}+\frac{\left(3 \sqrt{5}+15\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{30}+\frac{n^{3}}{6}-n^{2}+\\2^{-1+n} n +\frac{17 n}{6}-2^{n +1}+1 & \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) = 61
a(6) = 171
a(7) = 454
a(8) = 1167
a(n+4) = -4*a(n)+16*a(n+1)-17*a(n+2)+7*a(n+3)-1/6*(n-4)*(n-6)*(n+1), 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) = 20
a \! \left(5\right) = 61
a \! \left(6\right) = 171
a \! \left(7\right) = 454
a \! \left(8\right) = 1167
a \! \left(n +4\right) = -4 a \! \left(n \right)+16 a \! \left(n +1\right)-17 a \! \left(n +2\right)+7 a \! \left(n +3\right)-\frac{\left(n -4\right) \left(n -6\right) \left(n +1\right)}{6}, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15757/
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[29,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[16,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]+F[20,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[16,x]
F[12,x] = F[13,x]+F[17,x]
F[13,x] = F[1,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[13,x]*F[16,x]
F[16,x] = x
F[17,x] = F[10,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[16,x]*F[17,x]
F[20,x] = F[21,x]
F[21,x] = F[16,x]*F[22,x]
F[22,x] = F[23,x]+F[24,x]
F[23,x] = F[10,x]
F[24,x] = F[25,x]
F[25,x] = F[16,x]*F[26,x]
F[26,x] = F[27,x]+F[28,x]
F[27,x] = F[10,x]
F[28,x] = F[25,x]
F[29,x] = F[30,x]+F[41,x]
F[30,x] = F[31,x]
F[31,x] = F[16,x]*F[32,x]
F[32,x] = F[13,x]+F[33,x]
F[33,x] = F[30,x]+F[34,x]
F[34,x] = F[35,x]+F[36,x]+F[40,x]
F[35,x] = 0
F[36,x] = F[16,x]*F[37,x]
F[37,x] = F[38,x]+F[39,x]
F[38,x] = F[14,x]
F[39,x] = F[34,x]
F[40,x] = F[16,x]*F[33,x]
F[41,x] = F[35,x]+F[42,x]+F[64,x]
F[42,x] = F[16,x]*F[43,x]
F[43,x] = F[44,x]+F[48,x]
F[44,x] = F[45,x]+F[6,x]
F[45,x] = F[46,x]
F[46,x] = F[16,x]*F[47,x]
F[47,x] = F[6,x]
F[48,x] = F[41,x]+F[49,x]
F[49,x] = 2*F[35,x]+F[50,x]+F[54,x]
F[50,x] = F[16,x]*F[51,x]
F[51,x] = F[52,x]+F[53,x]
F[52,x] = F[45,x]
F[53,x] = F[49,x]
F[54,x] = F[16,x]*F[55,x]
F[55,x] = F[56,x]
F[56,x] = F[57,x]
F[57,x] = F[16,x]*F[58,x]
F[58,x] = F[44,x]+F[59,x]
F[59,x] = F[56,x]+F[60,x]
F[60,x] = 2*F[35,x]+F[54,x]+F[61,x]
F[61,x] = F[16,x]*F[62,x]
F[62,x] = F[52,x]+F[63,x]
F[63,x] = F[60,x]
F[64,x] = F[16,x]*F[65,x]
F[65,x] = F[66,x]+F[67,x]
F[66,x] = F[30,x]+F[56,x]
F[67,x] = F[68,x]
F[68,x] = F[35,x]+F[36,x]+F[69,x]
F[69,x] = F[16,x]*F[70,x]
F[70,x] = F[71,x]+F[76,x]
F[71,x] = F[30,x]+F[72,x]
F[72,x] = F[73,x]
F[73,x] = F[16,x]*F[74,x]
F[74,x] = F[38,x]+F[75,x]
F[75,x] = F[72,x]
F[76,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_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{29}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{16}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{5}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{20}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \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_{17}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{13}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = x
F_{17}\! \left(x \right) = F_{10}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{16}\! \left(x \right) F_{17}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{16}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \left(x \right)
F_{23}\! \left(x \right) = F_{10}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{16}\! \left(x \right) F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)
F_{27}\! \left(x \right) = F_{10}\! \left(x \right)
F_{28}\! \left(x \right) = F_{25}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{41}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{16}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{13}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{30}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{36}\! \left(x \right)+F_{40}\! \left(x \right)
F_{35}\! \left(x \right) = 0
F_{36}\! \left(x \right) = F_{16}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{39}\! \left(x \right)
F_{38}\! \left(x \right) = F_{14}\! \left(x \right)
F_{39}\! \left(x \right) = F_{34}\! \left(x \right)
F_{40}\! \left(x \right) = F_{16}\! \left(x \right) F_{33}\! \left(x \right)
F_{41}\! \left(x \right) = F_{35}\! \left(x \right)+F_{42}\! \left(x \right)+F_{64}\! \left(x \right)
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_{48}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{6}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{16}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{6}\! \left(x \right)
F_{48}\! \left(x \right) = F_{41}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = 2 F_{35}\! \left(x \right)+F_{50}\! \left(x \right)+F_{54}\! \left(x \right)
F_{50}\! \left(x \right) = F_{16}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{53}\! \left(x \right)
F_{52}\! \left(x \right) = F_{45}\! \left(x \right)
F_{53}\! \left(x \right) = F_{49}\! \left(x \right)
F_{54}\! \left(x \right) = F_{16}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{16}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{44}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{56}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = 2 F_{35}\! \left(x \right)+F_{54}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{16}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{52}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{60}\! \left(x \right)
F_{64}\! \left(x \right) = F_{16}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)+F_{67}\! \left(x \right)
F_{66}\! \left(x \right) = F_{30}\! \left(x \right)+F_{56}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{35}\! \left(x \right)+F_{36}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{16}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)+F_{76}\! \left(x \right)
F_{71}\! \left(x \right) = F_{30}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{16}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{38}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{72}\! \left(x \right)
F_{76}\! \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_16(x)*F_4(x))
Eq(F_4(x), F_29(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_16(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_20(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_16(x))
Eq(F_12(x), F_13(x) + F_17(x))
Eq(F_13(x), F_1(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_13(x)*F_16(x))
Eq(F_16(x), x)
Eq(F_17(x), F_10(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_16(x)*F_17(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_16(x)*F_22(x))
Eq(F_22(x), F_23(x) + F_24(x))
Eq(F_23(x), F_10(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_16(x)*F_26(x))
Eq(F_26(x), F_27(x) + F_28(x))
Eq(F_27(x), F_10(x))
Eq(F_28(x), F_25(x))
Eq(F_29(x), F_30(x) + F_41(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_16(x)*F_32(x))
Eq(F_32(x), F_13(x) + F_33(x))
Eq(F_33(x), F_30(x) + F_34(x))
Eq(F_34(x), F_35(x) + F_36(x) + F_40(x))
Eq(F_35(x), 0)
Eq(F_36(x), F_16(x)*F_37(x))
Eq(F_37(x), F_38(x) + F_39(x))
Eq(F_38(x), F_14(x))
Eq(F_39(x), F_34(x))
Eq(F_40(x), F_16(x)*F_33(x))
Eq(F_41(x), F_35(x) + F_42(x) + F_64(x))
Eq(F_42(x), F_16(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_48(x))
Eq(F_44(x), F_45(x) + F_6(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_16(x)*F_47(x))
Eq(F_47(x), F_6(x))
Eq(F_48(x), F_41(x) + F_49(x))
Eq(F_49(x), 2*F_35(x) + F_50(x) + F_54(x))
Eq(F_50(x), F_16(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_53(x))
Eq(F_52(x), F_45(x))
Eq(F_53(x), F_49(x))
Eq(F_54(x), F_16(x)*F_55(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_16(x)*F_58(x))
Eq(F_58(x), F_44(x) + F_59(x))
Eq(F_59(x), F_56(x) + F_60(x))
Eq(F_60(x), 2*F_35(x) + F_54(x) + F_61(x))
Eq(F_61(x), F_16(x)*F_62(x))
Eq(F_62(x), F_52(x) + F_63(x))
Eq(F_63(x), F_60(x))
Eq(F_64(x), F_16(x)*F_65(x))
Eq(F_65(x), F_66(x) + F_67(x))
Eq(F_66(x), F_30(x) + F_56(x))
Eq(F_67(x), F_68(x))
Eq(F_68(x), F_35(x) + F_36(x) + F_69(x))
Eq(F_69(x), F_16(x)*F_70(x))
Eq(F_70(x), F_71(x) + F_76(x))
Eq(F_71(x), F_30(x) + F_72(x))
Eq(F_72(x), F_73(x))
Eq(F_73(x), F_16(x)*F_74(x))
Eq(F_74(x), F_38(x) + F_75(x))
Eq(F_75(x), F_72(x))
Eq(F_76(x), F_68(x))
Pack JSON:
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Specification JSON:
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