0123_0231_1230_1302_2031
Counting sequence:
1, 1, 2, 6, 19, 56, 155, 413, 1076, 2763, 7022, 17705, 44356, 110532, 274181, 677419, 1667836, 4093490, 10018887, 24459644, 59578313, 144817933, 351341782, 850899545, 2057441706, 4967415357, 11976619172, 28839052998, 69359923065, 166629231509, 399889159158, 958742399566, 2296484931611, 5496014947664, 13142444580327, 31402765614317, 74979254531912, 178900846180135, 426576615822134, 1016504365268257, 2420818105001476, 5761930293743336, 13706865483163677, 32589914514765151, 77448519715105904, 183965811480284186, 436781164030375535, 1036577097441301268, 2458996408401742101, 5830958529641610173, 13821480318707506922, 32749758614756703925, 77572290102042584466, 183677426322073018037, 434771441735567994436, 1028791546252676292874, 2433659156582589546161, 5755230161081380845225, 13606316576282658614762, 32158628433887690377814, 75986680544240153823523, 179499889162122042075688, 423917381696954265724787, 1000903010498673550729725, 2362651940918211003676956, 5575805054905722086881667, 13155836303613724540937022, 31033818813401873681163993, 73191486953228562171570500, 172582639599856108342570764, 406862175837359005295783413, 958988909596754932988493971, 2259946695510556194349784036, 5324800543029500402968660674, 12543858263966946771344565527, 29544973229347992693327404012, 69576516054374752557206673697, 163821252482412330943531742765, 385663104235412077828166558014, 907774468121717060253115685137, 2136394862533395481391871568826, 5027133115077847902755239181421, 11827585482758138794915979002852, 27823396959246412151001944260046, 65443120233630040559942867557289, 153907231521424214363217614778941, 361906666291001137194544925197790, 850898910461803190051529939091998, 2000342842305670462074805336350251, 4701941314665628828959165140854208, 11050894827427357072451418959440447, 25969661995214902561186304329533645, 61021778172552084032941212944102576, 143368649993606360285201226810611167, 336802525793552979268827518836466566, 791132617303059142215509613099861329, 1858133939840566579633998777991327236, 4363745925972302169484519240701062960, 10247018544015619037101854419650858637, 24059804664371732268823371998206556359, 56486418422652594296652617057242763608
Generating function in Maple syntax:
-(x^3-x^2+2*x-1)*(2*x^4-6*x^3+9*x^2-5*x+1)/(x-1)^2/(x^3-2*x^2+3*x-1)^2
Generating function in latex syntax:
-\frac{\left(x^{3}-x^{2}+2 x -1\right) \left(2 x^{4}-6 x^{3}+9 x^{2}-5 x +1\right)}{\left(x -1\right)^{2} \left(x^{3}-2 x^{2}+3 x -1\right)^{2}}
Generating function in sympy syntax:
(-x**3 + x**2 - 2*x + 1)*(2*x**4 - 6*x**3 + 9*x**2 - 5*x + 1)/((x - 1)**2*(x**3 - 2*x**2 + 3*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(x-1)^2*(x^3-2*x^2+3*x-1)^2*F(x)+(x^3-x^2+2*x-1)*(2*x^4-6*x^3+9*x^2-5*x+1) = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right)^{2} \left(x^{3}-2 x^{2}+3 x -1\right)^{2} F \! \left(x \right)+\left(x^{3}-x^{2}+2 x -1\right) \left(2 x^{4}-6 x^{3}+9 x^{2}-5 x +1\right) = 0
Explicit closed form in Maple syntax:
1/317400*(-2760*(-1/4*(n+56/23)*(I-1/3*3^(1/2))*23^(1/2)+(n+3/4)*(I*3^(1/2)-1))*2^(2/3)*(11+3*23^(1/2)*3^(1/2))^(1/3)-345*2^(1/3)*((I+1/3*3^(1/2))*(n+1/115)*23^(1/2)+(1+I*3^(1/2))*(n+27/5))*(11+3*23^(1/2)*3^(1/2))^(2/3)+13800*n)*(11/600*((I-3/11*23^(1/2))*3^(1/2)-9/11*I*23^(1/2)+1)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3)-1/12*I*3^(1/2)*(44+12*23^(1/2)*3^(1/2))^(1/3)+1/12*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+1/317400*(2760*(-1/4*(n+56/23)*(I+1/3*3^(1/2))*23^(1/2)+(1+I*3^(1/2))*(n+3/4))*2^(2/3)*(11+3*23^(1/2)*3^(1/2))^(1/3)+345*((I-1/3*3^(1/2))*(n+1/115)*23^(1/2)+(n+27/5)*(I*3^(1/2)-1))*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3)+13800*n)*(-11/600*2^(1/3)*((I+3/11*23^(1/2))*3^(1/2)-9/11*I*23^(1/2)-1)*(11+3*23^(1/2)*3^(1/2))^(2/3)+1/12*I*3^(1/2)*(44+12*23^(1/2)*3^(1/2))^(1/3)+1/12*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+1/317400*(460*(3^(1/2)*(n+56/23)*23^(1/2)-12*n-9)*2^(2/3)*(11+3*23^(1/2)*3^(1/2))^(1/3)+230*2^(1/3)*(3^(1/2)*(n+1/115)*23^(1/2)+3*n+81/5)*(11+3*23^(1/2)*3^(1/2))^(2/3)+13800*n)*(1/300*2^(1/3)*(3*23^(1/2)*3^(1/2)-11)*(11+3*23^(1/2)*3^(1/2))^(2/3)-1/6*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)-n+1
Explicit closed form in latex syntax:
\frac{\left(-2760 \left(-\frac{\left(n +\frac{56}{23}\right) \left(\mathrm{I}-\frac{\sqrt{3}}{3}\right) \sqrt{23}}{4}+\left(n +\frac{3}{4}\right) \left(\mathrm{I} \sqrt{3}-1\right)\right) 2^{\frac{2}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}-345 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}+\frac{\sqrt{3}}{3}\right) \left(n +\frac{1}{115}\right) \sqrt{23}+\left(1+\mathrm{I} \sqrt{3}\right) \left(n +\frac{27}{5}\right)\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}+13800 n \right) \left(\frac{11 \left(\left(\mathrm{I}-\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}+1\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}-\frac{\mathrm{I} \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{317400}+\frac{\left(2760 \left(-\frac{\left(n +\frac{56}{23}\right) \left(\mathrm{I}+\frac{\sqrt{3}}{3}\right) \sqrt{23}}{4}+\left(1+\mathrm{I} \sqrt{3}\right) \left(n +\frac{3}{4}\right)\right) 2^{\frac{2}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+345 \left(\left(\mathrm{I}-\frac{\sqrt{3}}{3}\right) \left(n +\frac{1}{115}\right) \sqrt{23}+\left(n +\frac{27}{5}\right) \left(\mathrm{I} \sqrt{3}-1\right)\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}+13800 n \right) \left(-\frac{11 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}+\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}-1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}+\frac{\mathrm{I} \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{317400}+\frac{\left(460 \left(\sqrt{3}\, \left(n +\frac{56}{23}\right) \sqrt{23}-12 n -9\right) 2^{\frac{2}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+230 \,2^{\frac{1}{3}} \left(\sqrt{3}\, \left(n +\frac{1}{115}\right) \sqrt{23}+3 n +\frac{81}{5}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}+13800 n \right) \left(\frac{2^{\frac{1}{3}} \left(3 \sqrt{23}\, \sqrt{3}-11\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{300}-\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{2}{3}\right)^{-n}}{317400}-n +1
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 56
a(6) = 155
a(7) = 413
a(n+6) = -a(n)+4*a(n+1)-10*a(n+2)+14*a(n+3)-13*a(n+4)+6*a(n+5)-n-1, 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) = 56
a \! \left(6\right) = 155
a \! \left(7\right) = 413
a \! \left(n +6\right) = -a \! \left(n \right)+4 a \! \left(n +1\right)-10 a \! \left(n +2\right)+14 a \! \left(n +3\right)-13 a \! \left(n +4\right)+6 a \! \left(n +5\right)-n -1, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/17357/
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[19,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[16,x]*F[4,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[11,x]
F[18,x] = F[14,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[21,x]+F[22,x]+F[51,x]
F[21,x] = 0
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[24,x]+F[26,x]
F[24,x] = F[25,x]+F[7,x]
F[25,x] = F[15,x]
F[26,x] = F[20,x]+F[27,x]
F[27,x] = 2*F[21,x]+F[28,x]+F[32,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[30,x]+F[31,x]
F[30,x] = F[25,x]
F[31,x] = F[27,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[34,x]+F[45,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[39,x]
F[38,x] = F[11,x]+F[14,x]
F[39,x] = F[35,x]+F[40,x]
F[40,x] = 2*F[21,x]+F[32,x]+F[41,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[43,x]+F[44,x]
F[43,x] = F[14,x]
F[44,x] = F[40,x]
F[45,x] = F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]+F[50,x]
F[49,x] = F[35,x]
F[50,x] = F[46,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[53,x]+F[54,x]
F[53,x] = F[2,x]+F[35,x]
F[54,x] = F[46,x]+F[55,x]
F[55,x] = F[56,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[58,x]+F[59,x]
F[58,x] = F[2,x]
F[59,x] = F[55,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_{19}\! \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_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{11}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{51}\! \left(x \right)
F_{21}\! \left(x \right) = 0
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_{26}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{7}\! \left(x \right)
F_{25}\! \left(x \right) = F_{15}\! \left(x \right)
F_{26}\! \left(x \right) = F_{20}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{28}\! \left(x \right)+F_{32}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{31}\! \left(x \right)
F_{30}\! \left(x \right) = F_{25}\! \left(x \right)
F_{31}\! \left(x \right) = F_{27}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{45}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{39}\! \left(x \right)
F_{38}\! \left(x \right) = F_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{39}\! \left(x \right) = F_{35}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{32}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{44}\! \left(x \right)
F_{43}\! \left(x \right) = F_{14}\! \left(x \right)
F_{44}\! \left(x \right) = F_{40}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{50}\! \left(x \right)
F_{49}\! \left(x \right) = F_{35}\! \left(x \right)
F_{50}\! \left(x \right) = F_{46}\! \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_{54}\! \left(x \right)
F_{53}\! \left(x \right) = F_{2}\! \left(x \right)+F_{35}\! \left(x \right)
F_{54}\! \left(x \right) = F_{46}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{59}\! \left(x \right)
F_{58}\! \left(x \right) = F_{2}\! \left(x \right)
F_{59}\! \left(x \right) = F_{55}\! \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_19(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_16(x)*F_4(x))
Eq(F_16(x), F_17(x) + F_18(x))
Eq(F_17(x), F_11(x))
Eq(F_18(x), F_14(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_21(x) + F_22(x) + F_51(x))
Eq(F_21(x), 0)
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_24(x) + F_26(x))
Eq(F_24(x), F_25(x) + F_7(x))
Eq(F_25(x), F_15(x))
Eq(F_26(x), F_20(x) + F_27(x))
Eq(F_27(x), 2*F_21(x) + F_28(x) + F_32(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_30(x) + F_31(x))
Eq(F_30(x), F_25(x))
Eq(F_31(x), F_27(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_34(x) + F_45(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_39(x))
Eq(F_38(x), F_11(x) + F_14(x))
Eq(F_39(x), F_35(x) + F_40(x))
Eq(F_40(x), 2*F_21(x) + F_32(x) + F_41(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_44(x))
Eq(F_43(x), F_14(x))
Eq(F_44(x), F_40(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_50(x))
Eq(F_49(x), F_35(x))
Eq(F_50(x), F_46(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_53(x) + F_54(x))
Eq(F_53(x), F_2(x) + F_35(x))
Eq(F_54(x), F_46(x) + F_55(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_58(x) + F_59(x))
Eq(F_58(x), F_2(x))
Eq(F_59(x), F_55(x))
Pack JSON:
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Specification JSON:
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