0132_1203_1302_2130_2301
Counting sequence:
1, 1, 2, 6, 19, 54, 137, 319, 701, 1487, 3097, 6408, 13267, 27580, 57617, 120877, 254329, 535917, 1129617, 2379646, 5007071, 10519410, 22062601, 46190475, 96534773, 201407323, 419528681, 872533988, 1812081835, 3758266296, 7784829601, 16106364697, 33286274801, 68719801369, 141734297761, 292058212026, 601295923383, 1236951157006, 2542621297241, 5222680981335, 10720239221485, 21990233518119, 45079977824825, 92358977954944, 189116001348227, 387028094511092, 791648373711025, 1618481117993157, 3307330978468969, 6755399443404805, 13792273861421105, 28147497673935286, 57420895252136335, 117093590315111658, 238690780254456105, 486388759760200675, 990791918026091237, 2017612633066988595, 4107282860167353865, 8358680908405589724, 17005592192957463899, 34587645138212438128, 70328211781025290049, 142962266571257284913, 290536219160934374881, 590295810358715309361, 1199038364791131279169, 2434970217729672051826, 4943727411754171936103, 10035028776098009099014, 20365205457375358975609, 41320706725109410640175, 83822005070936218651869, 170005193383307244952447, 344732753249484119074649, 698910239464707511384824, 1416709944860893585219891, 2871198821584744312464684, 5817955506895402927313489, 11787026741242634479006813, 23876284937388926227732313, 48357032784585167017281437, 97922991388784963182072273, 198263834416799184684614382, 401363372112056886037276735, 812398150781030805439500130, 1644139114675895677639575113, 3326963855579459488832903995, 6731298963614255244807937237, 13617340432139183023936870923, 27544165874099711116554691113, 55707301767842112370512561364, 112652543574969605015875195403, 227780967228509970581496797736, 460513694614161462262535335393, 930930909542605966724205862473, 1881668859713778017846736731345, 3802951800684688204490181139273, 7685131763883640746573838470113, 15528719852795810168334693474602, 31374352355648677687043487624727
Generating function in Maple syntax:
(x^8-10*x^7+39*x^6-77*x^5+96*x^4-75*x^3+35*x^2-9*x+1)/(2*x-1)^2/(x-1)^6
Generating function in latex syntax:
\frac{x^{8}-10 x^{7}+39 x^{6}-77 x^{5}+96 x^{4}-75 x^{3}+35 x^{2}-9 x +1}{\left(2 x -1\right)^{2} \left(x -1\right)^{6}}
Generating function in sympy syntax:
(x**8 - 10*x**7 + 39*x**6 - 77*x**5 + 96*x**4 - 75*x**3 + 35*x**2 - 9*x + 1)/((x - 1)**6*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(2*x-1)^2*(x-1)^6*F(x)-x^8+10*x^7-39*x^6+77*x^5-96*x^4+75*x^3-35*x^2+9*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{2} \left(x -1\right)^{6} F \! \left(x \right)-x^{8}+10 x^{7}-39 x^{6}+77 x^{5}-96 x^{4}+75 x^{3}-35 x^{2}+9 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1+1/30*n+1/120*n^5-1/24*n^3+1/4*2^n*n-1/4*2^n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 1+\frac{n}{30}+\frac{n^{5}}{120}-\frac{n^{3}}{24}+\frac{2^{n} n}{4}-\frac{2^{n}}{4} & \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) = 54
a(6) = 137
a(7) = 319
a(8) = 701
a(n+2) = 1/120*n^5-1/12*n^4-1/24*n^3+7/12*n^2-4*a(n)+4*a(n+1)+8/15*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) = 19
a \! \left(5\right) = 54
a \! \left(6\right) = 137
a \! \left(7\right) = 319
a \! \left(8\right) = 701
a \! \left(n +2\right) = \frac{n^{5}}{120}-\frac{n^{4}}{12}-\frac{n^{3}}{24}+\frac{7 n^{2}}{12}-4 a \! \left(n \right)+4 a \! \left(n +1\right)+\frac{8 n}{15}+1, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19909/
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[36,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[25,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[17,x]+F[9,x]
F[17,x] = F[14,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[21,x]+F[24,x]
F[21,x] = F[22,x]
F[22,x] = F[12,x]*F[23,x]
F[23,x] = F[1,x]+F[21,x]
F[24,x] = F[19,x]
F[25,x] = F[26,x]+F[27,x]+F[31,x]
F[26,x] = 0
F[27,x] = F[12,x]*F[28,x]
F[28,x] = F[29,x]+F[30,x]
F[29,x] = F[10,x]
F[30,x] = F[25,x]
F[31,x] = F[12,x]*F[32,x]
F[32,x] = F[21,x]+F[33,x]
F[33,x] = F[26,x]+F[31,x]+F[34,x]
F[34,x] = F[12,x]*F[35,x]
F[35,x] = F[10,x]+F[33,x]
F[36,x] = F[37,x]+F[54,x]
F[37,x] = F[38,x]
F[38,x] = F[12,x]*F[39,x]
F[39,x] = F[40,x]+F[41,x]
F[40,x] = F[1,x]+F[14,x]
F[41,x] = F[37,x]+F[42,x]
F[42,x] = F[43,x]
F[43,x] = F[12,x]*F[44,x]
F[44,x] = F[20,x]+F[45,x]
F[45,x] = F[46,x]+F[47,x]
F[46,x] = F[43,x]
F[47,x] = F[48,x]
F[48,x] = F[12,x]*F[49,x]
F[49,x] = F[50,x]+F[53,x]
F[50,x] = F[51,x]
F[51,x] = F[12,x]*F[52,x]
F[52,x] = F[21,x]+F[50,x]
F[53,x] = F[48,x]
F[54,x] = F[26,x]+F[55,x]+F[59,x]
F[55,x] = F[12,x]*F[56,x]
F[56,x] = F[57,x]+F[58,x]
F[57,x] = F[6,x]
F[58,x] = F[54,x]
F[59,x] = F[12,x]*F[60,x]
F[60,x] = F[32,x]+F[61,x]
F[61,x] = F[62,x]+F[69,x]
F[62,x] = F[26,x]+F[63,x]+F[65,x]
F[63,x] = F[12,x]*F[64,x]
F[64,x] = F[14,x]+F[46,x]
F[65,x] = F[12,x]*F[66,x]
F[66,x] = F[20,x]+F[67,x]
F[67,x] = F[62,x]+F[68,x]
F[68,x] = F[48,x]
F[69,x] = F[26,x]+F[70,x]+F[71,x]+F[75,x]
F[70,x] = 0
F[71,x] = F[12,x]*F[72,x]
F[72,x] = F[73,x]+F[74,x]
F[73,x] = F[33,x]
F[74,x] = F[69,x]
F[75,x] = F[12,x]*F[76,x]
F[76,x] = F[50,x]+F[77,x]
F[77,x] = 2*F[26,x]+F[75,x]+F[78,x]
F[78,x] = F[12,x]*F[79,x]
F[79,x] = F[33,x]+F[77,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_{36}\! \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_{25}\! \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_{17}\! \left(x \right)+F_{9}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{24}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{1}\! \left(x \right)+F_{21}\! \left(x \right)
F_{24}\! \left(x \right) = F_{19}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{27}\! \left(x \right)+F_{31}\! \left(x \right)
F_{26}\! \left(x \right) = 0
F_{27}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{30}\! \left(x \right)
F_{29}\! \left(x \right) = F_{10}\! \left(x \right)
F_{30}\! \left(x \right) = F_{25}\! \left(x \right)
F_{31}\! \left(x \right) = F_{12}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{21}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{26}\! \left(x \right)+F_{31}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{10}\! \left(x \right)+F_{33}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{54}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{41}\! \left(x \right)
F_{40}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{41}\! \left(x \right) = F_{37}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{20}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{47}\! \left(x \right)
F_{46}\! \left(x \right) = F_{43}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{53}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{21}\! \left(x \right)+F_{50}\! \left(x \right)
F_{53}\! \left(x \right) = F_{48}\! \left(x \right)
F_{54}\! \left(x \right) = F_{26}\! \left(x \right)+F_{55}\! \left(x \right)+F_{59}\! \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_{58}\! \left(x \right)
F_{57}\! \left(x \right) = F_{6}\! \left(x \right)
F_{58}\! \left(x \right) = F_{54}\! \left(x \right)
F_{59}\! \left(x \right) = F_{12}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{32}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{69}\! \left(x \right)
F_{62}\! \left(x \right) = F_{26}\! \left(x \right)+F_{63}\! \left(x \right)+F_{65}\! \left(x \right)
F_{63}\! \left(x \right) = F_{12}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{14}\! \left(x \right)+F_{46}\! \left(x \right)
F_{65}\! \left(x \right) = F_{12}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{20}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{62}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{48}\! \left(x \right)
F_{69}\! \left(x \right) = F_{26}\! \left(x \right)+F_{70}\! \left(x \right)+F_{71}\! \left(x \right)+F_{75}\! \left(x \right)
F_{70}\! \left(x \right) = 0
F_{71}\! \left(x \right) = F_{12}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)+F_{74}\! \left(x \right)
F_{73}\! \left(x \right) = F_{33}\! \left(x \right)
F_{74}\! \left(x \right) = F_{69}\! \left(x \right)
F_{75}\! \left(x \right) = F_{12}\! \left(x \right) F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{50}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = 2 F_{26}\! \left(x \right)+F_{75}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{12}\! \left(x \right) F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{33}\! \left(x \right)+F_{77}\! \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_36(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_25(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_17(x) + F_9(x))
Eq(F_17(x), F_14(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_21(x) + F_24(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_12(x)*F_23(x))
Eq(F_23(x), F_1(x) + F_21(x))
Eq(F_24(x), F_19(x))
Eq(F_25(x), F_26(x) + F_27(x) + F_31(x))
Eq(F_26(x), 0)
Eq(F_27(x), F_12(x)*F_28(x))
Eq(F_28(x), F_29(x) + F_30(x))
Eq(F_29(x), F_10(x))
Eq(F_30(x), F_25(x))
Eq(F_31(x), F_12(x)*F_32(x))
Eq(F_32(x), F_21(x) + F_33(x))
Eq(F_33(x), F_26(x) + F_31(x) + F_34(x))
Eq(F_34(x), F_12(x)*F_35(x))
Eq(F_35(x), F_10(x) + F_33(x))
Eq(F_36(x), F_37(x) + F_54(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_12(x)*F_39(x))
Eq(F_39(x), F_40(x) + F_41(x))
Eq(F_40(x), F_1(x) + F_14(x))
Eq(F_41(x), F_37(x) + F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_12(x)*F_44(x))
Eq(F_44(x), F_20(x) + F_45(x))
Eq(F_45(x), F_46(x) + F_47(x))
Eq(F_46(x), F_43(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_12(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_53(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_12(x)*F_52(x))
Eq(F_52(x), F_21(x) + F_50(x))
Eq(F_53(x), F_48(x))
Eq(F_54(x), F_26(x) + F_55(x) + F_59(x))
Eq(F_55(x), F_12(x)*F_56(x))
Eq(F_56(x), F_57(x) + F_58(x))
Eq(F_57(x), F_6(x))
Eq(F_58(x), F_54(x))
Eq(F_59(x), F_12(x)*F_60(x))
Eq(F_60(x), F_32(x) + F_61(x))
Eq(F_61(x), F_62(x) + F_69(x))
Eq(F_62(x), F_26(x) + F_63(x) + F_65(x))
Eq(F_63(x), F_12(x)*F_64(x))
Eq(F_64(x), F_14(x) + F_46(x))
Eq(F_65(x), F_12(x)*F_66(x))
Eq(F_66(x), F_20(x) + F_67(x))
Eq(F_67(x), F_62(x) + F_68(x))
Eq(F_68(x), F_48(x))
Eq(F_69(x), F_26(x) + F_70(x) + F_71(x) + F_75(x))
Eq(F_70(x), 0)
Eq(F_71(x), F_12(x)*F_72(x))
Eq(F_72(x), F_73(x) + F_74(x))
Eq(F_73(x), F_33(x))
Eq(F_74(x), F_69(x))
Eq(F_75(x), F_12(x)*F_76(x))
Eq(F_76(x), F_50(x) + F_77(x))
Eq(F_77(x), 2*F_26(x) + F_75(x) + F_78(x))
Eq(F_78(x), F_12(x)*F_79(x))
Eq(F_79(x), F_33(x) + F_77(x))
Pack JSON:
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Specification JSON:
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