0132_1032_1203_2130_2301
Counting sequence:
1, 1, 2, 6, 19, 50, 115, 244, 496, 991, 1978, 3979, 8093, 16636, 34485, 71894, 150382, 315021, 660036, 1382053, 2890691, 6037938, 12593287, 26226776, 54540604, 113263435, 234901150, 486562639, 1006659961, 2080405816, 4295002793, 8858410474, 18253656858, 37581015641, 77309469640, 158913855369, 326417587647, 670014979726, 1374389625371, 2817498646668, 5772436156936, 11819750121143, 24189255945922, 49478023397971, 101155069917589, 206708186199220, 422212465259485, 862017116387134, 1759218604670726, 3588805953309541, 7318349394746508, 14918173765956269, 30399297485065723, 61924494876683946, 126100789566739695, 256705178760511744, 522417556775400212, 1062849512059890531, 2161727821138324006, 4395513236314124183, 8935141660703620081, 18158513697558433648, 36893488147419736657, 74939897799445728466, 152185638608104519474, 308982963234635753905, 627189298506125567120, 1272825341085959923729, 2582544170319338140791, 5238875316933513628166, 10625324586456702757123, 21545797078092757373364, 43681889966544219374688, 88544371553805848969551, 179449926349046519400650, 363622219180962682805019, 736689171327664654760077, 1492267808586807889027436, 3022314549036572938343941, 6120186961799060198610598, 12391489651049949042484126, 25085210757003555376987453, 50774884423814425339585492, 102758694667243479852046261, 207935240973716218051582259, 420706185225890952799971490, 851083777008698938995476055, 1721510367131231944784032424, 3481706360490132023156338156, 7040783973435600313491437691, 14236310451781873161342718638, 28782105913385091391407553759, 58183181846412872920261883753, 117604303732111126115419980456, 237684487542793012780635168505, 480320735242727546660863659194, 970544990799738135520916999242, 1960897022228042355440216530441, 3961408125713216879677201433176, 8002044413940698096947943061913, 16162545152909924869082970113071
Generating function in Maple syntax:
-(3*x^8-13*x^7+24*x^6-33*x^5+48*x^4-48*x^3+27*x^2-8*x+1)/(2*x-1)^2/(x-1)^5
Generating function in latex syntax:
-\frac{3 x^{8}-13 x^{7}+24 x^{6}-33 x^{5}+48 x^{4}-48 x^{3}+27 x^{2}-8 x +1}{\left(2 x -1\right)^{2} \left(x -1\right)^{5}}
Generating function in sympy syntax:
(-3*x**8 + 13*x**7 - 24*x**6 + 33*x**5 - 48*x**4 + 48*x**3 - 27*x**2 + 8*x - 1)/((x - 1)**5*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(2*x-1)^2*(x-1)^5*F(x)+3*x^8-13*x^7+24*x^6-33*x^5+48*x^4-48*x^3+27*x^2-8*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{2} \left(x -1\right)^{5} F \! \left(x \right)+3 x^{8}-13 x^{7}+24 x^{6}-33 x^{5}+48 x^{4}-48 x^{3}+27 x^{2}-8 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/24*(3*n+6)*2^n+1/24*n^4+1/12*n^3-13/24*n^2-7/12*n+2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(3 n +6\right) 2^{n}}{24}+\frac{n^{4}}{24}+\frac{n^{3}}{12}-\frac{13 n^{2}}{24}-\frac{7 n}{12}+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) = 50
a(6) = 115
a(7) = 244
a(8) = 496
a(n+2) = 1/24*n^4-1/4*n^3-25/24*n^2-4*a(n)+4*a(n+1)+9/4*n+4, 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) = 50
a \! \left(6\right) = 115
a \! \left(7\right) = 244
a \! \left(8\right) = 496
a \! \left(n +2\right) = \frac{n^{4}}{24}-\frac{n^{3}}{4}-\frac{25 n^{2}}{24}-4 a \! \left(n \right)+4 a \! \left(n +1\right)+\frac{9 n}{4}+4, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19701/
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[64,x]
F[55,x] = F[12,x]*F[56,x]
F[56,x] = F[57,x]+F[58,x]
F[57,x] = F[10,x]+F[18,x]
F[58,x] = F[59,x]+F[60,x]
F[59,x] = F[26,x]+F[31,x]+F[55,x]
F[60,x] = F[61,x]
F[61,x] = F[12,x]*F[62,x]
F[62,x] = F[50,x]+F[63,x]
F[63,x] = F[61,x]
F[64,x] = F[12,x]*F[65,x]
F[65,x] = F[32,x]+F[66,x]
F[66,x] = F[67,x]+F[74,x]
F[67,x] = F[26,x]+F[68,x]+F[70,x]
F[68,x] = F[12,x]*F[69,x]
F[69,x] = F[10,x]+F[24,x]
F[70,x] = F[12,x]*F[71,x]
F[71,x] = F[20,x]+F[72,x]
F[72,x] = F[67,x]+F[73,x]
F[73,x] = F[48,x]
F[74,x] = F[26,x]+F[75,x]+F[76,x]+F[80,x]
F[75,x] = 0
F[76,x] = F[12,x]*F[77,x]
F[77,x] = F[78,x]+F[79,x]
F[78,x] = F[33,x]
F[79,x] = F[74,x]
F[80,x] = F[12,x]*F[81,x]
F[81,x] = F[50,x]+F[82,x]
F[82,x] = 2*F[26,x]+F[80,x]+F[83,x]
F[83,x] = F[12,x]*F[84,x]
F[84,x] = F[33,x]+F[82,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_{64}\! \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_{10}\! \left(x \right)+F_{18}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{60}\! \left(x \right)
F_{59}\! \left(x \right) = F_{26}\! \left(x \right)+F_{31}\! \left(x \right)+F_{55}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{12}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{50}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{61}\! \left(x \right)
F_{64}\! \left(x \right) = F_{12}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{32}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)+F_{74}\! \left(x \right)
F_{67}\! \left(x \right) = F_{26}\! \left(x \right)+F_{68}\! \left(x \right)+F_{70}\! \left(x \right)
F_{68}\! \left(x \right) = F_{12}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{10}\! \left(x \right)+F_{24}\! \left(x \right)
F_{70}\! \left(x \right) = F_{12}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{20}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{67}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{48}\! \left(x \right)
F_{74}\! \left(x \right) = F_{26}\! \left(x \right)+F_{75}\! \left(x \right)+F_{76}\! \left(x \right)+F_{80}\! \left(x \right)
F_{75}\! \left(x \right) = 0
F_{76}\! \left(x \right) = F_{12}\! \left(x \right) F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{78}\! \left(x \right)+F_{79}\! \left(x \right)
F_{78}\! \left(x \right) = F_{33}\! \left(x \right)
F_{79}\! \left(x \right) = F_{74}\! \left(x \right)
F_{80}\! \left(x \right) = F_{12}\! \left(x \right) F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{50}\! \left(x \right)+F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = 2 F_{26}\! \left(x \right)+F_{80}\! \left(x \right)+F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{12}\! \left(x \right) F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{33}\! \left(x \right)+F_{82}\! \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_64(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_10(x) + F_18(x))
Eq(F_58(x), F_59(x) + F_60(x))
Eq(F_59(x), F_26(x) + F_31(x) + F_55(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_12(x)*F_62(x))
Eq(F_62(x), F_50(x) + F_63(x))
Eq(F_63(x), F_61(x))
Eq(F_64(x), F_12(x)*F_65(x))
Eq(F_65(x), F_32(x) + F_66(x))
Eq(F_66(x), F_67(x) + F_74(x))
Eq(F_67(x), F_26(x) + F_68(x) + F_70(x))
Eq(F_68(x), F_12(x)*F_69(x))
Eq(F_69(x), F_10(x) + F_24(x))
Eq(F_70(x), F_12(x)*F_71(x))
Eq(F_71(x), F_20(x) + F_72(x))
Eq(F_72(x), F_67(x) + F_73(x))
Eq(F_73(x), F_48(x))
Eq(F_74(x), F_26(x) + F_75(x) + F_76(x) + F_80(x))
Eq(F_75(x), 0)
Eq(F_76(x), F_12(x)*F_77(x))
Eq(F_77(x), F_78(x) + F_79(x))
Eq(F_78(x), F_33(x))
Eq(F_79(x), F_74(x))
Eq(F_80(x), F_12(x)*F_81(x))
Eq(F_81(x), F_50(x) + F_82(x))
Eq(F_82(x), 2*F_26(x) + F_80(x) + F_83(x))
Eq(F_83(x), F_12(x)*F_84(x))
Eq(F_84(x), F_33(x) + F_82(x))
Pack JSON:
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Specification JSON:
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