0132_0213_1320_3102
Counting sequence:
1, 1, 2, 6, 20, 63, 182, 495, 1303, 3382, 8745, 22632, 58710, 152667, 397762, 1037817, 2710515, 7084046, 18523235, 48450256, 126758598, 331690153, 868044322, 2271911861, 5946634545, 15565884778, 40746813687, 106666153650, 279234853498, 731004833017, 1913712514100, 5009998465397, 13116014416505, 34337507878672, 89895435438287, 235346650907782, 616140222267120, 1613065425901901, 4223038875505280, 11056016840803791, 28944942927349253, 75778674502201368, 198390805701249017, 519393192845622366, 1359787673323869520, 3559967628102597643, 9320110812937266192, 24400356014616018555, 63880939638724569843, 167242427717185410852, 437846273144087276605, 1146296250977587836912, 3001042198313699277300, 7856829781013556306827, 20569446018827062511762, 53851506023667817231147, 140985067548576761474475, 369103687614863212088506, 966325977281614364918725, 2529874208201182863284276, 6623296575264340186556058, 17340015373476649620045583, 45396749256935232521352462, 118850231820868295640036021, 311153945052748149791318695, 814611601031533144519596658, 2132680853430165265339412447, 5583430950035590614643150460, 14617611978229862504879727338, 38269404947760508752576120637, 100190602791264687458009570510, 262302403278459601031775268177, 686716606748966210457962443845, 1797847416378143219983405390382, 4706825641204871828774841343187, 12322629504875289024906294889662, 32261062868698628763074396904830, 84460559091775864298577604122861, 221120614387739498201179833264320, 578901284033663698442004732542961, 1515583237637693733398920039452013, 3967848428728301774302926737402548, 10387962048244980134606202877424907, 27196037715402175719708367305779826, 71200151096752621204904269863380820, 186404415572437836255775183933031073, 488013095615725184283962765234873452, 1277634871265066310039196078371827115, 3344891518160130932719791403083094233, 8757039683176640861892509997284537372, 22926227531292420400502402321586892565, 60021642910545877834704024433111206442, 157138701200035728093788325909019285140, 411394460688942336427018263156494305087, 1077044680865553341147981083285561598812, 2819739581905241806938354226150389206327, 7382174064845220319509940074066006352935, 19326782612620515631277182953848433550216, 50598173772996519533693042703080904859065, 132467738706329428888544812986597505453596, 346805042345912538969427131919118063801368
Generating function in Maple syntax:
(x^9-5*x^8+2*x^7+24*x^6-60*x^5+82*x^4-69*x^3+34*x^2-9*x+1)/(2*x-1)/(x^2-3*x+1)/(x-1)^5
Generating function in latex syntax:
\frac{x^{9}-5 x^{8}+2 x^{7}+24 x^{6}-60 x^{5}+82 x^{4}-69 x^{3}+34 x^{2}-9 x +1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{5}}
Generating function in sympy syntax:
(x**9 - 5*x**8 + 2*x**7 + 24*x**6 - 60*x**5 + 82*x**4 - 69*x**3 + 34*x**2 - 9*x + 1)/((x - 1)**5*(2*x - 1)*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
-(2*x-1)*(x^2-3*x+1)*(x-1)^5*F(x)+x^9-5*x^8+2*x^7+24*x^6-60*x^5+82*x^4-69*x^3+34*x^2-9*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{5} F \! \left(x \right)+x^{9}-5 x^{8}+2 x^{7}+24 x^{6}-60 x^{5}+82 x^{4}-69 x^{3}+34 x^{2}-9 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/120*(-24*5^(1/2)+120)*(3/2-1/2*5^(1/2))^(-n)+1/120*(24*5^(1/2)+120)*(3/2+1/2*5^(1/2))^(-n)+1/24*n^4-1/4*n^3-13/24*n^2+7/4*n+1/4*2^n-3)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(-24 \sqrt{5}+120\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{120}+\frac{\left(24 \sqrt{5}+120\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{120}+\frac{n^{4}}{24}-\frac{n^{3}}{4}-\\\frac{13 n^{2}}{24}+\frac{7 n}{4}+\frac{2^{n}}{4}-3 & \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(6) = 182
a(7) = 495
a(8) = 1303
a(9) = 3382
a(n+3) = 1/24*n^4-1/4*n^3-37/24*n^2+2*a(n)-7*a(n+1)+5*a(n+2)+15/4*n+1, n >= 10
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(6\right) = 182
a \! \left(7\right) = 495
a \! \left(8\right) = 1303
a \! \left(9\right) = 3382
a \! \left(n +3\right) = \frac{n^{4}}{24}-\frac{n^{3}}{4}-\frac{37 n^{2}}{24}+2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)+\frac{15 n}{4}+1, \quad n \geq 10
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15293/
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[16,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[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[14,x]+F[7,x]
F[14,x] = F[15,x]
F[15,x] = F[13,x]*F[4,x]
F[16,x] = F[17,x]+F[27,x]
F[17,x] = F[18,x]
F[18,x] = F[19,x]*F[4,x]
F[19,x] = F[20,x]+F[6,x]
F[20,x] = F[17,x]+F[21,x]
F[21,x] = F[22,x]
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[24,x]
F[24,x] = F[17,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[24,x]*F[4,x]
F[27,x] = F[28,x]+F[29,x]+F[65,x]
F[28,x] = 0
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[31,x]+F[49,x]
F[31,x] = F[32,x]+F[7,x]
F[32,x] = F[28,x]+F[33,x]+F[41,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[35,x]+F[38,x]
F[35,x] = F[11,x]+F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[35,x]*F[4,x]
F[38,x] = F[14,x]+F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[38,x]*F[4,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[13,x]+F[43,x]
F[43,x] = F[44,x]+F[47,x]
F[44,x] = F[45,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[35,x]
F[47,x] = F[48,x]
F[48,x] = F[4,x]*F[43,x]
F[49,x] = F[27,x]+F[50,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[54,x]+F[63,x]
F[54,x] = F[28,x]+F[55,x]+F[61,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[57,x]+F[59,x]
F[57,x] = F[11,x]+F[58,x]
F[58,x] = F[15,x]+F[28,x]+F[33,x]
F[59,x] = F[54,x]+F[60,x]
F[60,x] = F[51,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[17,x]+F[54,x]
F[63,x] = F[64,x]
F[64,x] = F[4,x]*F[53,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[62,x]+F[67,x]
F[67,x] = F[68,x]+F[72,x]
F[68,x] = F[69,x]
F[69,x] = F[4,x]*F[70,x]
F[70,x] = F[71,x]
F[71,x] = F[44,x]+F[7,x]
F[72,x] = 2*F[28,x]+F[73,x]+F[77,x]
F[73,x] = F[4,x]*F[74,x]
F[74,x] = F[75,x]
F[75,x] = F[14,x]+F[76,x]
F[76,x] = F[48,x]
F[77,x] = F[4,x]*F[67,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_{16}\! \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_{13}\! \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_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{27}\! \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_{6}\! \left(x \right)
F_{20}\! \left(x \right) = F_{17}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{17}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{29}\! \left(x \right)+F_{65}\! \left(x \right)
F_{28}\! \left(x \right) = 0
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{49}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{7}\! \left(x \right)
F_{32}\! \left(x \right) = F_{28}\! \left(x \right)+F_{33}\! \left(x \right)+F_{41}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{38}\! \left(x \right)
F_{35}\! \left(x \right) = F_{11}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{14}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{13}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{47}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{35}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{49}\! \left(x \right) = F_{27}\! \left(x \right)+F_{50}\! \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_{54}\! \left(x \right)+F_{63}\! \left(x \right)
F_{54}\! \left(x \right) = F_{28}\! \left(x \right)+F_{55}\! \left(x \right)+F_{61}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{59}\! \left(x \right)
F_{57}\! \left(x \right) = F_{11}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{15}\! \left(x \right)+F_{28}\! \left(x \right)+F_{33}\! \left(x \right)
F_{59}\! \left(x \right) = F_{54}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{51}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{17}\! \left(x \right)+F_{54}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{62}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)+F_{72}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{4}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{44}\! \left(x \right)+F_{7}\! \left(x \right)
F_{72}\! \left(x \right) = 2 F_{28}\! \left(x \right)+F_{73}\! \left(x \right)+F_{77}\! \left(x \right)
F_{73}\! \left(x \right) = F_{4}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{14}\! \left(x \right)+F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{48}\! \left(x \right)
F_{77}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \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_16(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_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_7(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_13(x)*F_4(x))
Eq(F_16(x), F_17(x) + F_27(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_6(x))
Eq(F_20(x), F_17(x) + F_21(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_17(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_24(x)*F_4(x))
Eq(F_27(x), F_28(x) + F_29(x) + F_65(x))
Eq(F_28(x), 0)
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_31(x) + F_49(x))
Eq(F_31(x), F_32(x) + F_7(x))
Eq(F_32(x), F_28(x) + F_33(x) + F_41(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_35(x) + F_38(x))
Eq(F_35(x), F_11(x) + F_36(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_35(x)*F_4(x))
Eq(F_38(x), F_14(x) + F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_38(x)*F_4(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_13(x) + F_43(x))
Eq(F_43(x), F_44(x) + F_47(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_35(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_4(x)*F_43(x))
Eq(F_49(x), F_27(x) + F_50(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_54(x) + F_63(x))
Eq(F_54(x), F_28(x) + F_55(x) + F_61(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_57(x) + F_59(x))
Eq(F_57(x), F_11(x) + F_58(x))
Eq(F_58(x), F_15(x) + F_28(x) + F_33(x))
Eq(F_59(x), F_54(x) + F_60(x))
Eq(F_60(x), F_51(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_17(x) + F_54(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_4(x)*F_53(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_62(x) + F_67(x))
Eq(F_67(x), F_68(x) + F_72(x))
Eq(F_68(x), F_69(x))
Eq(F_69(x), F_4(x)*F_70(x))
Eq(F_70(x), F_71(x))
Eq(F_71(x), F_44(x) + F_7(x))
Eq(F_72(x), 2*F_28(x) + F_73(x) + F_77(x))
Eq(F_73(x), F_4(x)*F_74(x))
Eq(F_74(x), F_75(x))
Eq(F_75(x), F_14(x) + F_76(x))
Eq(F_76(x), F_48(x))
Eq(F_77(x), F_4(x)*F_67(x))
Pack JSON:
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Specification JSON:
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