0132_0231_1203_2130_3120
Counting sequence:
1, 1, 2, 6, 19, 55, 149, 381, 930, 2191, 5027, 11309, 25068, 54943, 119365, 257501, 552294, 1178847, 2505799, 5307293, 11205344, 23591439, 49543465, 103807021, 217052954, 452982255, 943715499, 1962931021, 4076859860, 8455712831, 17515409037, 36238781629, 74893486798, 154618816703, 318901315215, 657129989181, 1352914690504, 2783138799407, 5720896429169, 11751030512013, 24120536323714, 49478023238479, 101429947650035, 207807697636461, 425510999935164, 870813209183455, 1781208836980949, 3641582511176861, 7441494696769590, 15199648742355871, 31032616182329047, 63331869759875549, 129197014310167728, 263460578201149263, 537054255563905465, 1094374709451002861, 2229281815548366314, 4539628424389429167, 9241386435364225339, 18807032043899157133, 38262582434139698084, 77822201560962133119, 158238476507289707805, 321665099785310264701, 653706493112082191774, 1328165573307087670655, 2697836320780021875999, 5478682989891736779901, 11123386676446859572120, 22578814746220491123311, 45821712279094526157057, 92971590131496140085069, 188599511409606455659858, 382511685112441262244623, 775648694811339226282115, 1572548038795591856090541, 3187597375937010519171724, 6460197348565674652260127, 13090399890514656532286309, 26520810167795927520034653, 53721641109125083950920454, 108803323765316625723467359, 220326730624766167090108775, 446093627437798165466483741, 903067587252127993505414784, 1827895839257319312155635855, 3699313008020765274600792649, 7485668675053783849780532141, 15145422668132074300718859450, 30639015972313161803753207151, 61974373216724350012137285067, 125341428977644752833536202189, 253468223043681611285595555188, 512507176264147433808237294783, 1036155812881863290090566837165, 2094594546470863425129318044221, 4233754934356000540155004698734, 8556641551540548460102746484287, 17291546468738191679790967004079, 34939619668790572878752881936573, 70592292800209524795847659582824
Generating function in Maple syntax:
(3*x^8-7*x^7+5*x^6-8*x^5+20*x^4-28*x^3+20*x^2-7*x+1)/(2*x-1)^2/(x-1)^4
Generating function in latex syntax:
\frac{3 x^{8}-7 x^{7}+5 x^{6}-8 x^{5}+20 x^{4}-28 x^{3}+20 x^{2}-7 x +1}{\left(2 x -1\right)^{2} \left(x -1\right)^{4}}
Generating function in sympy syntax:
(3*x**8 - 7*x**7 + 5*x**6 - 8*x**5 + 20*x**4 - 28*x**3 + 20*x**2 - 7*x + 1)/((x - 1)**4*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
-(2*x-1)^2*(x-1)^4*F(x)+3*x^8-7*x^7+5*x^6-8*x^5+20*x^4-28*x^3+20*x^2-7*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right)^{2} \left(x -1\right)^{4} F \! \left(x \right)+3 x^{8}-7 x^{7}+5 x^{6}-8 x^{5}+20 x^{4}-28 x^{3}+20 x^{2}-7 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1/48*(27*n-27)*2^n-1/6*n^3+7/6*n-2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(27 n -27\right) 2^{n}}{48}-\frac{n^{3}}{6}+\frac{7 n}{6}-2 & \mathit{\text{otherwise}} \end{array}\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) = 149
a(7) = 381
a(8) = 930
a(n+2) = -4*a(n)+4*a(n+1)-1/6*(-2+n)*(n^2-4*n-15), 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) = 149
a \! \left(7\right) = 381
a \! \left(8\right) = 930
a \! \left(n +2\right) = -4 a \! \left(n \right)+4 a \! \left(n +1\right)-\frac{\left(-2+n \right) \left(n^{2}-4 n -15\right)}{6}, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18996/
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[12,x]*F[4,x]
F[4,x] = F[22,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[17,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[1,x]+F[14,x]
F[17,x] = F[18,x]+F[19,x]+F[21,x]
F[18,x] = 0
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[10,x]+F[17,x]
F[21,x] = F[12,x]*F[13,x]
F[22,x] = F[23,x]+F[32,x]
F[23,x] = F[24,x]
F[24,x] = F[12,x]*F[25,x]
F[25,x] = F[26,x]+F[31,x]
F[26,x] = F[1,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[12,x]*F[29,x]
F[29,x] = F[30,x]+F[9,x]
F[30,x] = F[14,x]
F[31,x] = F[23,x]
F[32,x] = F[18,x]+F[33,x]+F[51,x]
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[35,x]+F[50,x]
F[35,x] = F[36,x]+F[6,x]
F[36,x] = F[37,x]
F[37,x] = F[12,x]*F[38,x]
F[38,x] = F[39,x]+F[42,x]
F[39,x] = F[14,x]+F[40,x]
F[40,x] = F[41,x]
F[41,x] = F[12,x]*F[39,x]
F[42,x] = F[43,x]+F[46,x]
F[43,x] = F[44,x]
F[44,x] = F[12,x]*F[45,x]
F[45,x] = F[14,x]+F[43,x]
F[46,x] = 2*F[18,x]+F[47,x]+F[49,x]
F[47,x] = F[12,x]*F[48,x]
F[48,x] = F[40,x]+F[46,x]
F[49,x] = F[12,x]*F[42,x]
F[50,x] = F[32,x]
F[51,x] = F[12,x]*F[52,x]
F[52,x] = F[53,x]+F[61,x]
F[53,x] = F[23,x]+F[54,x]
F[54,x] = F[18,x]+F[55,x]+F[60,x]
F[55,x] = F[12,x]*F[56,x]
F[56,x] = F[57,x]+F[59,x]
F[57,x] = F[10,x]+F[58,x]
F[58,x] = F[41,x]
F[59,x] = F[54,x]
F[60,x] = F[12,x]*F[53,x]
F[61,x] = F[62,x]+F[69,x]
F[62,x] = F[18,x]+F[63,x]+F[67,x]
F[63,x] = F[12,x]*F[64,x]
F[64,x] = F[65,x]
F[65,x] = F[14,x]+F[66,x]
F[66,x] = F[44,x]
F[67,x] = F[12,x]*F[68,x]
F[68,x] = F[23,x]+F[62,x]
F[69,x] = F[18,x]+F[70,x]+F[77,x]+F[79,x]
F[70,x] = F[12,x]*F[71,x]
F[71,x] = F[72,x]
F[72,x] = F[40,x]+F[73,x]
F[73,x] = F[74,x]
F[74,x] = F[12,x]*F[75,x]
F[75,x] = F[43,x]+F[76,x]
F[76,x] = F[74,x]
F[77,x] = F[12,x]*F[78,x]
F[78,x] = F[54,x]+F[69,x]
F[79,x] = F[12,x]*F[61,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{22}\! \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_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{17}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{10}\! \left(x \right)+F_{17}\! \left(x \right)
F_{21}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{32}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{31}\! \left(x \right)
F_{26}\! \left(x \right) = F_{1}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{9}\! \left(x \right)
F_{30}\! \left(x \right) = F_{14}\! \left(x \right)
F_{31}\! \left(x \right) = F_{23}\! \left(x \right)
F_{32}\! \left(x \right) = F_{18}\! \left(x \right)+F_{33}\! \left(x \right)+F_{51}\! \left(x \right)
F_{33}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{50}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{6}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{42}\! \left(x \right)
F_{39}\! \left(x \right) = F_{14}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{46}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{14}\! \left(x \right)+F_{43}\! \left(x \right)
F_{46}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{47}\! \left(x \right)+F_{49}\! \left(x \right)
F_{47}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{40}\! \left(x \right)+F_{46}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right) F_{42}\! \left(x \right)
F_{50}\! \left(x \right) = F_{32}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{61}\! \left(x \right)
F_{53}\! \left(x \right) = F_{23}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{18}\! \left(x \right)+F_{55}\! \left(x \right)+F_{60}\! \left(x \right)
F_{55}\! \left(x \right) = F_{12}\! \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_{10}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{41}\! \left(x \right)
F_{59}\! \left(x \right) = F_{54}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{53}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{69}\! \left(x \right)
F_{62}\! \left(x \right) = F_{18}\! \left(x \right)+F_{63}\! \left(x \right)+F_{67}\! \left(x \right)
F_{63}\! \left(x \right) = F_{12}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{14}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{44}\! \left(x \right)
F_{67}\! \left(x \right) = F_{12}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{23}\! \left(x \right)+F_{62}\! \left(x \right)
F_{69}\! \left(x \right) = F_{18}\! \left(x \right)+F_{70}\! \left(x \right)+F_{77}\! \left(x \right)+F_{79}\! \left(x \right)
F_{70}\! \left(x \right) = F_{12}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{40}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{12}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{43}\! \left(x \right)+F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{74}\! \left(x \right)
F_{77}\! \left(x \right) = F_{12}\! \left(x \right) F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{54}\! \left(x \right)+F_{69}\! \left(x \right)
F_{79}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_22(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_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_17(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_1(x) + F_14(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_21(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_10(x) + F_17(x))
Eq(F_21(x), F_12(x)*F_13(x))
Eq(F_22(x), F_23(x) + F_32(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_12(x)*F_25(x))
Eq(F_25(x), F_26(x) + F_31(x))
Eq(F_26(x), F_1(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_12(x)*F_29(x))
Eq(F_29(x), F_30(x) + F_9(x))
Eq(F_30(x), F_14(x))
Eq(F_31(x), F_23(x))
Eq(F_32(x), F_18(x) + F_33(x) + F_51(x))
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_35(x) + F_50(x))
Eq(F_35(x), F_36(x) + F_6(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_12(x)*F_38(x))
Eq(F_38(x), F_39(x) + F_42(x))
Eq(F_39(x), F_14(x) + F_40(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_12(x)*F_39(x))
Eq(F_42(x), F_43(x) + F_46(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_12(x)*F_45(x))
Eq(F_45(x), F_14(x) + F_43(x))
Eq(F_46(x), 2*F_18(x) + F_47(x) + F_49(x))
Eq(F_47(x), F_12(x)*F_48(x))
Eq(F_48(x), F_40(x) + F_46(x))
Eq(F_49(x), F_12(x)*F_42(x))
Eq(F_50(x), F_32(x))
Eq(F_51(x), F_12(x)*F_52(x))
Eq(F_52(x), F_53(x) + F_61(x))
Eq(F_53(x), F_23(x) + F_54(x))
Eq(F_54(x), F_18(x) + F_55(x) + F_60(x))
Eq(F_55(x), F_12(x)*F_56(x))
Eq(F_56(x), F_57(x) + F_59(x))
Eq(F_57(x), F_10(x) + F_58(x))
Eq(F_58(x), F_41(x))
Eq(F_59(x), F_54(x))
Eq(F_60(x), F_12(x)*F_53(x))
Eq(F_61(x), F_62(x) + F_69(x))
Eq(F_62(x), F_18(x) + F_63(x) + F_67(x))
Eq(F_63(x), F_12(x)*F_64(x))
Eq(F_64(x), F_65(x))
Eq(F_65(x), F_14(x) + F_66(x))
Eq(F_66(x), F_44(x))
Eq(F_67(x), F_12(x)*F_68(x))
Eq(F_68(x), F_23(x) + F_62(x))
Eq(F_69(x), F_18(x) + F_70(x) + F_77(x) + F_79(x))
Eq(F_70(x), F_12(x)*F_71(x))
Eq(F_71(x), F_72(x))
Eq(F_72(x), F_40(x) + F_73(x))
Eq(F_73(x), F_74(x))
Eq(F_74(x), F_12(x)*F_75(x))
Eq(F_75(x), F_43(x) + F_76(x))
Eq(F_76(x), F_74(x))
Eq(F_77(x), F_12(x)*F_78(x))
Eq(F_78(x), F_54(x) + F_69(x))
Eq(F_79(x), F_12(x)*F_61(x))
Pack JSON:
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Specification JSON:
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