0132_0321_1203_1230_3102
Counting sequence:
1, 1, 2, 6, 19, 48, 99, 187, 340, 607, 1080, 1928, 3461, 6250, 11345, 20677, 37798, 69241, 127022, 233242, 428551, 787716, 1448255, 2663103, 4897480, 9007059, 16565668, 30468028, 56038361, 103069438, 189572973, 348677673, 641316730, 1179563757, 2169554266, 3990430574, 7339544123, 13499524184, 24829493787, 45668556675, 83997568892, 154495613255, 284161732368, 522654907696, 961312246125, 1768128878610, 3252096024457, 5981537140813, 11001762035086, 20235395191137, 37218694357382, 68455851573506, 125909941111471, 231584487031340, 425950279704823, 783444707835655, 1440979474559344, 2650374462086843, 4874798644468348, 8966152581100516, 16491325687641153, 30332276913194918, 55789755181920933, 102613357782740785, 188735389877839842, 347138502842484181, 638487250503046834, 1174361143223352278, 2159986896568864099, 3972835290295243392, 7307183330087439315, 13440005516951525707, 24720024137334186660, 45467212984373129263, 83627242638658818536, 153814479760366110680, 282908935383398034005, 520350657782422938042, 957074072926187056833, 1760333666092008002261, 3237758396800617969782, 5955166135818813000777, 10953258198711438943966, 20146182731330869884906, 37054607065861121799255, 68154047995903430596948, 125354837793095422249135, 230563492854859974612559, 424072378643858827425048, 779990709291814224252323, 1434626580790533026254676, 2638689668726206077895948, 4853306958808553328365993, 8926623208325292432478798, 16418619835860051838702045, 30198550002993897599507257, 55543793047179241870647626, 102160962886033191308815549, 187903305936206330778928138, 345608061869418763958348094, 635672330691658286046047627
Generating function in Maple syntax:
-(4*x^7+9*x^6+3*x^5-4*x^4-x^3-3*x^2+3*x-1)/(x^3+x^2+x-1)/(x-1)^3
Generating function in latex syntax:
-\frac{4 x^{7}+9 x^{6}+3 x^{5}-4 x^{4}-x^{3}-3 x^{2}+3 x -1}{\left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(-4*x**7 - 9*x**6 - 3*x**5 + 4*x**4 + x**3 + 3*x**2 - 3*x + 1)/((x - 1)**3*(x**3 + x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(x^3+x^2+x-1)*(x-1)^3*F(x)+4*x^7+9*x^6+3*x^5-4*x^4-x^3-3*x^2+3*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+4 x^{7}+9 x^{6}+3 x^{5}-4 x^{4}-x^{3}-3 x^{2}+3 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,n = 1,1,1/528*(((-29*11^(1/2)+143*I)*3^(1/2)-87*I*11^(1/2)+143)*(17+3*11^(1/2)*3^(1/2))^(1/3)+616+((110*I+16*11^(1/2))*3^(1/2)-48*I*11^(1/2)-110)*(17+3*11^(1/2)*3^(1/2))^(2/3))*(1/24*((17*I+3*11^(1/2))*3^(1/2)-9*I*11^(1/2)-17)*(17+3*11^(1/2)*3^(1/2))^(2/3)-1/6*I*3^(1/2)*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/6*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/3)^(-n)+1/528*(((-110*I+16*11^(1/2))*3^(1/2)+48*I*11^(1/2)-110)*(17+3*11^(1/2)*3^(1/2))^(2/3)+616+((-143*I-29*11^(1/2))*3^(1/2)+87*I*11^(1/2)+143)*(17+3*11^(1/2)*3^(1/2))^(1/3))*(1/24*((-17*I+3*11^(1/2))*3^(1/2)+9*I*11^(1/2)-17)*(17+3*11^(1/2)*3^(1/2))^(2/3)+1/6*I*3^(1/2)*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/6*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/3)^(-n)+1/528*((-32*11^(1/2)*3^(1/2)+220)*(17+3*11^(1/2)*3^(1/2))^(2/3)+58*(17+3*11^(1/2)*3^(1/2))^(1/3)*11^(1/2)*3^(1/2)-286*(17+3*11^(1/2)*3^(1/2))^(1/3)+616)*(1/3*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/3+17/12*(17+3*11^(1/2)*3^(1/2))^(2/3)-1/4*(17+3*11^(1/2)*3^(1/2))^(2/3)*11^(1/2)*3^(1/2))^(-n)+5/2*n^2-15*n+29/2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ \frac{\left(\left(\left(-29 \sqrt{11}+143 \,\mathrm{I}\right) \sqrt{3}-87 \,\mathrm{I} \sqrt{11}+143\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+616+\left(\left(110 \,\mathrm{I}+16 \sqrt{11}\right) \sqrt{3}-48 \,\mathrm{I} \sqrt{11}-110\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}-9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}-\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{528}\\+\\\frac{\left(\left(\left(-110 \,\mathrm{I}+16 \sqrt{11}\right) \sqrt{3}+48 \,\mathrm{I} \sqrt{11}-110\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+616+\left(\left(-143 \,\mathrm{I}-29 \sqrt{11}\right) \sqrt{3}+87 \,\mathrm{I} \sqrt{11}+143\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{\left(\left(-17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}+9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}+\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{528}\\+\\\frac{\left(\left(-32 \sqrt{11}\, \sqrt{3}+220\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+58 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{11}\, \sqrt{3}-286 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+616\right) \left(\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}-\frac{1}{3}+\frac{17 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{12}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{11}\, \sqrt{3}}{4}\right)^{-n}}{528}\\+\frac{5 n^{2}}{2}-15 n +\frac{29}{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) = 48
a(6) = 99
a(7) = 187
a(n+3) = -5*n^2+a(n)+a(n+1)+a(n+2)+30*n-19, n >= 8
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) = 48
a \! \left(6\right) = 99
a \! \left(7\right) = 187
a \! \left(n +3\right) = -5 n^{2}+a \! \left(n \right)+a \! \left(n +1\right)+a \! \left(n +2\right)+30 n -19, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19401/
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[21,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[11,x]+F[14,x]
F[14,x] = F[15,x]+F[16,x]+F[19,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[11,x]+F[18,x]
F[18,x] = F[16,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[11,x]
F[21,x] = F[22,x]+F[39,x]
F[22,x] = F[23,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[31,x]
F[25,x] = F[1,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[10,x]+F[29,x]
F[29,x] = F[11,x]+F[30,x]
F[30,x] = F[19,x]
F[31,x] = F[22,x]+F[32,x]
F[32,x] = F[33,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[35,x]+F[36,x]
F[35,x] = F[22,x]
F[36,x] = F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[35,x]*F[4,x]
F[39,x] = F[15,x]+F[40,x]+F[55,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]+F[50,x]
F[42,x] = F[43,x]+F[7,x]
F[43,x] = F[15,x]+F[44,x]+F[47,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[17,x]+F[46,x]
F[46,x] = F[14,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[20,x]+F[49,x]
F[49,x] = F[18,x]
F[50,x] = F[39,x]+F[51,x]
F[51,x] = F[52,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[54,x]
F[54,x] = F[37,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[35,x]+F[57,x]
F[57,x] = F[58,x]
F[58,x] = F[15,x]+F[38,x]+F[59,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[61,x]+F[66,x]
F[61,x] = F[11,x]+F[62,x]
F[62,x] = F[15,x]+F[19,x]+F[63,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[17,x]+F[65,x]
F[65,x] = F[30,x]
F[66,x] = F[37,x]+F[67,x]
F[67,x] = F[68,x]
F[68,x] = F[4,x]*F[54,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_{21}\! \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_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{19}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{11}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{16}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{11}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{39}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{31}\! \left(x \right)
F_{25}\! \left(x \right) = F_{1}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{10}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{11}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{19}\! \left(x \right)
F_{31}\! \left(x \right) = F_{22}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \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_{36}\! \left(x \right)
F_{35}\! \left(x \right) = F_{22}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{15}\! \left(x \right)+F_{40}\! \left(x \right)+F_{55}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{50}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{7}\! \left(x \right)
F_{43}\! \left(x \right) = F_{15}\! \left(x \right)+F_{44}\! \left(x \right)+F_{47}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{17}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{14}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{20}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{18}\! \left(x \right)
F_{50}\! \left(x \right) = F_{39}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{37}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{35}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{15}\! \left(x \right)+F_{38}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)+F_{66}\! \left(x \right)
F_{61}\! \left(x \right) = F_{11}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{15}\! \left(x \right)+F_{19}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{17}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{30}\! \left(x \right)
F_{66}\! \left(x \right) = F_{37}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \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_21(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_11(x) + F_14(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_19(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_11(x) + F_18(x))
Eq(F_18(x), F_16(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_11(x))
Eq(F_21(x), F_22(x) + F_39(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_31(x))
Eq(F_25(x), F_1(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_10(x) + F_29(x))
Eq(F_29(x), F_11(x) + F_30(x))
Eq(F_30(x), F_19(x))
Eq(F_31(x), F_22(x) + F_32(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_35(x) + F_36(x))
Eq(F_35(x), F_22(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_35(x)*F_4(x))
Eq(F_39(x), F_15(x) + F_40(x) + F_55(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_50(x))
Eq(F_42(x), F_43(x) + F_7(x))
Eq(F_43(x), F_15(x) + F_44(x) + F_47(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_17(x) + F_46(x))
Eq(F_46(x), F_14(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_20(x) + F_49(x))
Eq(F_49(x), F_18(x))
Eq(F_50(x), F_39(x) + F_51(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_37(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_35(x) + F_57(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_15(x) + F_38(x) + F_59(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_61(x) + F_66(x))
Eq(F_61(x), F_11(x) + F_62(x))
Eq(F_62(x), F_15(x) + F_19(x) + F_63(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_17(x) + F_65(x))
Eq(F_65(x), F_30(x))
Eq(F_66(x), F_37(x) + F_67(x))
Eq(F_67(x), F_68(x))
Eq(F_68(x), F_4(x)*F_54(x))
Pack JSON:
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Specification JSON:
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