0123_0231_0312_1203_2130
Counting sequence:
1, 1, 2, 6, 19, 54, 143, 360, 871, 2046, 4706, 10667, 23940, 53378, 118514, 262437, 580191, 1281411, 2828471, 6241179, 13768803, 30372271, 66993266, 147764442, 325912056, 718830327, 1585440370, 3496810696, 7712472571, 17010409668, 37517657871, 82747820242, 182506086607, 402529872532, 887807612242, 1958121364045, 4318772660156, 9525352999264, 21008827437090, 46336427617327, 102198208326087, 225405244191353, 497146916112823, 1096492040676045, 2418389325680139, 5333925567623085, 11764343176086434, 25947075678032452, 57228076923883696, 126220497024066877, 278388069726397730, 614004216376930330, 1354228929778199587, 2986845929283091106, 6587696074943430223, 14529621079665402572, 32046088088614265079, 70679872252172356986, 155889365584010542466, 343824819256635806847, 758329510765444460084, 1672548387114899986318, 3688921593486436339218, 8136172697738317736137, 17944893782591536095983, 39578709158669509210303, 87293591015077336879607, 192532075812746210623887, 424642860784161931274739, 936579312583401200295931, 2065690700979548612135058, 4556024262743259156518974, 10048627838069919514365224, 22162946377119387641956563, 48881917016982034441585426, 107812461872033988398754300, 237787870121187364440750987, 524457657259356763324443600, 1156727776390747515049070927, 2551243422902682394540398422, 5626944503064721552406825183, 12410616782520190619864388272, 27372476987943063634270927346, 60371898478950848820950520897, 133154413740421888261767363052, 293681304468786840157807681804, 647734507416524529136568011714, 1428623428573470946534905616115, 3150928161615728733227621249751, 6949590830647981995591812956757, 15327805089869434937718534088823, 33806538341354598608664692104161, 74562667513357179212921199963419, 164453140116583793363560936939673, 362712818574522185335786569037378, 799988304662401549884494341226184, 1764429749441386893132549622718560, 3891572317457295971600885817943993, 8583132939576993493086265980731202, 18930695628595373879305081587950190, 41752963574648043730211048997770547
Generating function in Maple syntax:
(x^8-5*x^7+10*x^6-12*x^5+16*x^4-19*x^3+15*x^2-6*x+1)/(x^3+2*x-1)/(x-1)^5
Generating function in latex syntax:
\frac{x^{8}-5 x^{7}+10 x^{6}-12 x^{5}+16 x^{4}-19 x^{3}+15 x^{2}-6 x +1}{\left(x^{3}+2 x -1\right) \left(x -1\right)^{5}}
Generating function in sympy syntax:
(x**8 - 5*x**7 + 10*x**6 - 12*x**5 + 16*x**4 - 19*x**3 + 15*x**2 - 6*x + 1)/((x - 1)**5*(x**3 + 2*x - 1))
Implicit equation for the generating function in Maple syntax:
(x^3+2*x-1)*(x-1)^5*F(x)-x^8+5*x^7-10*x^6+12*x^5-16*x^4+19*x^3-15*x^2+6*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{3}+2 x -1\right) \left(x -1\right)^{5} F \! \left(x \right)-x^{8}+5 x^{7}-10 x^{6}+12 x^{5}-16 x^{4}+19 x^{3}-15 x^{2}+6 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n < 0,473/30208*(108+12*59^(1/2)*3^(1/2))^(-2/3*n)*((-3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)+9*3^(5/6)*(I+1/9*59^(1/2))*(36+4*59^(1/2)*3^(1/2))^(1/3)-48*I*3^(1/2)-48)^(-n)*(-(1/331776*(3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)-1/36864*3^(5/6)*(I-1/9*59^(1/2))*(36+4*59^(1/2)*3^(1/2))^(1/3)+1/6912*I*3^(1/2)-1/6912)^(-n)*((1888/1419*n^4+7552/1419*n^3+37760/1419*n^2+139712/1419*n+42480/473)*(-1/12*(108+12*59^(1/2)*3^(1/2))^(1/3)-1/12*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+1/96*((-I*59^(1/2)-3)*18^(1/3)+9*I*2^(1/3)*3^(1/6)+3^(1/6)*2^(1/3)*59^(1/2))*(9+59^(1/2)*3^(1/2))^(2/3))^n+((I*3^(1/3)+1/3*3^(5/6))*59^(1/2)-413/1419*I*3^(5/6)-413/1419*3^(1/3))*2^(2/3)*(9+59^(1/2)*3^(1/2))^(1/3)-14160/473+961/2838*2^(1/3)*((I*3^(2/3)-3^(1/6))*59^(1/2)-20001/961*I*3^(1/6)+6667/961*3^(2/3))*(9+59^(1/2)*3^(1/2))^(2/3))*(-1/12*(108+12*59^(1/2)*3^(1/2))^(1/3)+1/12*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+1/96*((I*59^(1/2)-3)*18^(1/3)-9*I*3^(1/6)*2^(1/3)+3^(1/6)*2^(1/3)*59^(1/2))*(9+59^(1/2)*3^(1/2))^(2/3))^n+(((-1/3*3^(5/6)+I*3^(1/3))*59^(1/2)-413/1419*I*3^(5/6)+413/1419*3^(1/3))*2^(2/3)*(9+59^(1/2)*3^(1/2))^(1/3)+14160/473+961/2838*2^(1/3)*((I*3^(2/3)+3^(1/6))*59^(1/2)-20001/961*I*3^(1/6)-6667/961*3^(2/3))*(9+59^(1/2)*3^(1/2))^(2/3))*(1/331776*(3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)-1/36864*3^(5/6)*(I-1/9*59^(1/2))*(36+4*59^(1/2)*3^(1/2))^(1/3)+1/6912*I*3^(1/2)-1/6912)^(-n)*(-1/12*(108+12*59^(1/2)*3^(1/2))^(1/3)-1/12*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+1/96*((-I*59^(1/2)-3)*18^(1/3)+9*I*2^(1/3)*3^(1/6)+3^(1/6)*2^(1/3)*59^(1/2))*(9+59^(1/2)*3^(1/2))^(2/3))^n+2/3*((-I*59^(1/2)+3)*(81+9*59^(1/2)*3^(1/2))^(1/3)+9*3^(1/6)*(I-1/9*59^(1/2))*(9+59^(1/2)*3^(1/2))^(1/3)-8*I*3^(5/6)*2^(1/3)+8*6^(1/3))^n*(1/(59^(1/2)*3^(1/6)*(9+59^(1/2)*3^(1/2))^(1/3)-3*(81+9*59^(1/2)*3^(1/2))^(1/3)-8*6^(1/3)))^n*((2^(2/3)*(59^(1/2)*3^(5/6)-413/473*3^(1/3))*(9+59^(1/2)*3^(1/2))^(1/3)-961/946*2^(1/3)*(59^(1/2)*3^(1/6)-6667/961*3^(2/3))*(9+59^(1/2)*3^(1/2))^(2/3))*((3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)-9*3^(5/6)*(I-1/9*59^(1/2))*(36+4*59^(1/2)*3^(1/2))^(1/3)+48*I*3^(1/2)-48)^(-n)*(-13824*(108+12*59^(1/2)*3^(1/2))^(1/3)-13824*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+((-1728*I*59^(1/2)-5184)*18^(1/3)+15552*I*2^(1/3)*3^(1/6)+1728*3^(1/6)*2^(1/3)*59^(1/2))*(9+59^(1/2)*3^(1/2))^(2/3))^n+21240/473*(-27648*(108+12*59^(1/2)*3^(1/2))^(1/3)-27648*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+((-3456*I*59^(1/2)-10368)*18^(1/3)+31104*I*3^(1/6)*2^(1/3)+3456*3^(1/6)*2^(1/3)*59^(1/2))*(9+59^(1/2)*3^(1/2))^(2/3))^n*((6*I*59^(1/2)-18)*(108+12*59^(1/2)*3^(1/2))^(1/3)-18*3^(5/6)*(I-1/9*59^(1/2))*(36+4*59^(1/2)*3^(1/2))^(1/3)+96*I*3^(1/2)-96)^(-n))),n = 0,1,0 < n,473/30208*(108+12*59^(1/2)*3^(1/2))^(-2/3*n)*(-(1/331776*(3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)-1/36864*3^(5/6)*(I-1/9*59^(1/2))*(36+4*59^(1/2)*3^(1/2))^(1/3)+1/6912*I*3^(1/2)-1/6912)^(-n)*((1888/1419*n^4+7552/1419*n^3+37760/1419*n^2+139712/1419*n+42480/473)*(-1/12*(108+12*59^(1/2)*3^(1/2))^(1/3)-1/12*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+1/96*((-I*59^(1/2)-3)*18^(1/3)+9*I*2^(1/3)*3^(1/6)+3^(1/6)*2^(1/3)*59^(1/2))*(9+59^(1/2)*3^(1/2))^(2/3))^n+((I*3^(1/3)+1/3*3^(5/6))*59^(1/2)-413/1419*I*3^(5/6)-413/1419*3^(1/3))*2^(2/3)*(9+59^(1/2)*3^(1/2))^(1/3)-14160/473+961/2838*2^(1/3)*((I*3^(2/3)-3^(1/6))*59^(1/2)-20001/961*I*3^(1/6)+6667/961*3^(2/3))*(9+59^(1/2)*3^(1/2))^(2/3))*((-3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)+9*3^(5/6)*(I+1/9*59^(1/2))*(36+4*59^(1/2)*3^(1/2))^(1/3)-48*I*3^(1/2)-48)^(-n)*(-1/12*(108+12*59^(1/2)*3^(1/2))^(1/3)+1/12*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+1/96*((I*59^(1/2)-3)*18^(1/3)-9*I*3^(1/6)*2^(1/3)+3^(1/6)*2^(1/3)*59^(1/2))*(9+59^(1/2)*3^(1/2))^(2/3))^n+(1/331776*(3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)-1/36864*3^(5/6)*(I-1/9*59^(1/2))*(36+4*59^(1/2)*3^(1/2))^(1/3)+1/6912*I*3^(1/2)-1/6912)^(-n)*((-3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)+9*3^(5/6)*(I+1/9*59^(1/2))*(36+4*59^(1/2)*3^(1/2))^(1/3)-48*I*3^(1/2)-48)^(-n)*(((-1/3*3^(5/6)+I*3^(1/3))*59^(1/2)-413/1419*I*3^(5/6)+413/1419*3^(1/3))*2^(2/3)*(9+59^(1/2)*3^(1/2))^(1/3)+14160/473+961/2838*2^(1/3)*((I*3^(2/3)+3^(1/6))*59^(1/2)-20001/961*I*3^(1/6)-6667/961*3^(2/3))*(9+59^(1/2)*3^(1/2))^(2/3))*(-1/12*(108+12*59^(1/2)*3^(1/2))^(1/3)-1/12*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+1/96*((-I*59^(1/2)-3)*18^(1/3)+9*I*2^(1/3)*3^(1/6)+3^(1/6)*2^(1/3)*59^(1/2))*(9+59^(1/2)*3^(1/2))^(2/3))^n-961/1419*((-3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)+9*3^(5/6)*(I+1/9*59^(1/2))*(36+4*59^(1/2)*3^(1/2))^(1/3)-48*I*3^(1/2)-48)^(-n)*59^(1/2)*3^(1/6)*2^(1/3)*(9+59^(1/2)*3^(1/2))^(2/3)*(-59^(1/2)*3^(1/6)*(9+59^(1/2)*3^(1/2))^(1/3)+3*(81+9*59^(1/2)*3^(1/2))^(1/3)+8*6^(1/3))^(-n)*((3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)-9*3^(5/6)*(I-1/9*59^(1/2))*(36+4*59^(1/2)*3^(1/2))^(1/3)+48*I*3^(1/2)-48)^(-n)*(18+2*59^(1/2)*3^(1/2))^(-1/3*n)*(-8*(108+12*59^(1/2)*3^(1/2))^(1/3)+8*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+((I*59^(1/2)-3)*18^(1/3)-9*I*3^(1/6)*2^(1/3)+3^(1/6)*2^(1/3)*59^(1/2))*(9+59^(1/2)*3^(1/2))^(2/3))^n*(-13824*(108+12*59^(1/2)*3^(1/2))^(1/3)-13824*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+((-1728*I*59^(1/2)-5184)*18^(1/3)+15552*I*2^(1/3)*3^(1/6)+1728*3^(1/6)*2^(1/3)*59^(1/2))*(9+59^(1/2)*3^(1/2))^(2/3))^n+14160/473*((-3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)+9*3^(5/6)*(I+1/9*59^(1/2))*(36+4*59^(1/2)*3^(1/2))^(1/3)-48*I*3^(1/2)-48)^(-n)*(1/(59^(1/2)*3^(1/6)*(9+59^(1/2)*3^(1/2))^(1/3)-3*(81+9*59^(1/2)*3^(1/2))^(1/3)-8*6^(1/3)))^n*(1/165888*(3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)-1/18432*3^(5/6)*(I-1/9*59^(1/2))*(36+4*59^(1/2)*3^(1/2))^(1/3)+1/3456*I*3^(1/2)-1/3456)^(-n)*(1/96*(-I*59^(1/2)+3)*(81+9*59^(1/2)*3^(1/2))^(1/3)+3/32*3^(1/6)*(I-1/9*59^(1/2))*(9+59^(1/2)*3^(1/2))^(1/3)-1/12*I*3^(5/6)*2^(1/3)+1/12*6^(1/3))^n*(-8*(108+12*59^(1/2)*3^(1/2))^(1/3)-8*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+((-I*59^(1/2)-3)*18^(1/3)+9*I*2^(1/3)*3^(1/6)+3^(1/6)*2^(1/3)*59^(1/2))*(9+59^(1/2)*3^(1/2))^(2/3))^n+2/3*(2^(2/3)*(59^(1/2)*3^(5/6)-413/473*3^(1/3))*(9+59^(1/2)*3^(1/2))^(1/3)+6667/946*2^(1/3)*3^(2/3)*(9+59^(1/2)*3^(1/2))^(2/3))*(-59^(1/2)*3^(1/6)*(9+59^(1/2)*3^(1/2))^(1/3)+3*(81+9*59^(1/2)*3^(1/2))^(1/3)+8*6^(1/3))^(-n)*(18+2*59^(1/2)*3^(1/2))^(1/3*n)*((3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)-9*3^(5/6)*(I-1/9*59^(1/2))*(36+4*59^(1/2)*3^(1/2))^(1/3)+48*I*3^(1/2)-48)^(-n)*((I*59^(1/2)-3)*(81+9*59^(1/2)*3^(1/2))^(1/3)-9*3^(1/6)*(I-1/9*59^(1/2))*(9+59^(1/2)*3^(1/2))^(1/3)+8*I*3^(5/6)*2^(1/3)-8*6^(1/3))^n*((-I*59^(1/2)-3)*(81+9*59^(1/2)*3^(1/2))^(1/3)+9*3^(1/6)*(I+1/9*59^(1/2))*(9+59^(1/2)*3^(1/2))^(1/3)-8*I*3^(5/6)*2^(1/3)-8*6^(1/3))^n*(1/1728*(-3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)+1/192*3^(5/6)*(I+1/9*59^(1/2))*(36+4*59^(1/2)*3^(1/2))^(1/3)-1/36*I*3^(1/2)-1/36)^(-n)))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}\frac{473 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}} \left(\left(-3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+9 \,3^{\frac{5}{6}} \left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-48 \,\mathrm{I} \sqrt{3}-48\right)^{-n} \left(-\left(\frac{\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{331776}-\frac{3^{\frac{5}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{36864}+\frac{\mathrm{I} \sqrt{3}}{6912}-\frac{1}{6912}\right)^{-n} \left(\left(\frac{1888}{1419} n^{4}+\frac{7552}{1419} n^{3}+\frac{37760}{1419} n^{2}+\frac{139712}{1419} n +\frac{42480}{473}\right) \left(-\frac{\left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{12}+\frac{\left(\left(-\mathrm{I} \sqrt{59}-3\right) 18^{\frac{1}{3}}+9 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+3^{\frac{1}{6}} 2^{\frac{1}{3}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{96}\right)^{n}+\left(\left(\mathrm{I} \,3^{\frac{1}{3}}+\frac{3^{\frac{5}{6}}}{3}\right) \sqrt{59}-\frac{413 \,\mathrm{I} \,3^{\frac{5}{6}}}{1419}-\frac{413 \,3^{\frac{1}{3}}}{1419}\right) 2^{\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{14160}{473}+\frac{961 \,2^{\frac{1}{3}} \left(\left(\mathrm{I} \,3^{\frac{2}{3}}-3^{\frac{1}{6}}\right) \sqrt{59}-\frac{20001 \,\mathrm{I} \,3^{\frac{1}{6}}}{961}+\frac{6667 \,3^{\frac{2}{3}}}{961}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{2838}\right) \left(-\frac{\left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{12}+\frac{\left(\left(\mathrm{I} \sqrt{59}-3\right) 18^{\frac{1}{3}}-9 \,\mathrm{I} \,3^{\frac{1}{6}} 2^{\frac{1}{3}}+3^{\frac{1}{6}} 2^{\frac{1}{3}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{96}\right)^{n}+\left(\left(\left(-\frac{3^{\frac{5}{6}}}{3}+\mathrm{I} \,3^{\frac{1}{3}}\right) \sqrt{59}-\frac{413 \,\mathrm{I} \,3^{\frac{5}{6}}}{1419}+\frac{413 \,3^{\frac{1}{3}}}{1419}\right) 2^{\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{14160}{473}+\frac{961 \,2^{\frac{1}{3}} \left(\left(\mathrm{I} \,3^{\frac{2}{3}}+3^{\frac{1}{6}}\right) \sqrt{59}-\frac{20001 \,\mathrm{I} \,3^{\frac{1}{6}}}{961}-\frac{6667 \,3^{\frac{2}{3}}}{961}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{2838}\right) \left(\frac{\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{331776}-\frac{3^{\frac{5}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{36864}+\frac{\mathrm{I} \sqrt{3}}{6912}-\frac{1}{6912}\right)^{-n} \left(-\frac{\left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{12}+\frac{\left(\left(-\mathrm{I} \sqrt{59}-3\right) 18^{\frac{1}{3}}+9 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+3^{\frac{1}{6}} 2^{\frac{1}{3}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{96}\right)^{n}+\frac{2 \left(\left(-\mathrm{I} \sqrt{59}+3\right) \left(81+9 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+9 \,3^{\frac{1}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-8 \,\mathrm{I} \,3^{\frac{5}{6}} 2^{\frac{1}{3}}+8 \,6^{\frac{1}{3}}\right)^{n} \left(\frac{1}{\sqrt{59}\, 3^{\frac{1}{6}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-3 \left(81+9 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-8 \,6^{\frac{1}{3}}}\right)^{n} \left(\left(2^{\frac{2}{3}} \left(\sqrt{59}\, 3^{\frac{5}{6}}-\frac{413 \,3^{\frac{1}{3}}}{473}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{961 \,2^{\frac{1}{3}} \left(\sqrt{59}\, 3^{\frac{1}{6}}-\frac{6667 \,3^{\frac{2}{3}}}{961}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{946}\right) \left(\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-9 \,3^{\frac{5}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+48 \,\mathrm{I} \sqrt{3}-48\right)^{-n} \left(-13824 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-13824 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\left(\left(-1728 \,\mathrm{I} \sqrt{59}-5184\right) 18^{\frac{1}{3}}+15552 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+1728 \,3^{\frac{1}{6}} 2^{\frac{1}{3}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n}+\frac{21240 \left(-27648 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-27648 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\left(\left(-3456 \,\mathrm{I} \sqrt{59}-10368\right) 18^{\frac{1}{3}}+31104 \,\mathrm{I} \,3^{\frac{1}{6}} 2^{\frac{1}{3}}+3456 \,3^{\frac{1}{6}} 2^{\frac{1}{3}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n} \left(\left(6 \,\mathrm{I} \sqrt{59}-18\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-18 \,3^{\frac{5}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+96 \,\mathrm{I} \sqrt{3}-96\right)^{-n}}{473}\right)}{3}\right)}{30208} & n <0 \\ 1 & n =0 \\ \frac{473 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}} \left(-\left(\frac{\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{331776}-\frac{3^{\frac{5}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{36864}+\frac{\mathrm{I} \sqrt{3}}{6912}-\frac{1}{6912}\right)^{-n} \left(\left(\frac{1888}{1419} n^{4}+\frac{7552}{1419} n^{3}+\frac{37760}{1419} n^{2}+\frac{139712}{1419} n +\frac{42480}{473}\right) \left(-\frac{\left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{12}+\frac{\left(\left(-\mathrm{I} \sqrt{59}-3\right) 18^{\frac{1}{3}}+9 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+3^{\frac{1}{6}} 2^{\frac{1}{3}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{96}\right)^{n}+\left(\left(\mathrm{I} \,3^{\frac{1}{3}}+\frac{3^{\frac{5}{6}}}{3}\right) \sqrt{59}-\frac{413 \,\mathrm{I} \,3^{\frac{5}{6}}}{1419}-\frac{413 \,3^{\frac{1}{3}}}{1419}\right) 2^{\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{14160}{473}+\frac{961 \,2^{\frac{1}{3}} \left(\left(\mathrm{I} \,3^{\frac{2}{3}}-3^{\frac{1}{6}}\right) \sqrt{59}-\frac{20001 \,\mathrm{I} \,3^{\frac{1}{6}}}{961}+\frac{6667 \,3^{\frac{2}{3}}}{961}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{2838}\right) \left(\left(-3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+9 \,3^{\frac{5}{6}} \left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-48 \,\mathrm{I} \sqrt{3}-48\right)^{-n} \left(-\frac{\left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{12}+\frac{\left(\left(\mathrm{I} \sqrt{59}-3\right) 18^{\frac{1}{3}}-9 \,\mathrm{I} \,3^{\frac{1}{6}} 2^{\frac{1}{3}}+3^{\frac{1}{6}} 2^{\frac{1}{3}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{96}\right)^{n}+\left(\frac{\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{331776}-\frac{3^{\frac{5}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{36864}+\frac{\mathrm{I} \sqrt{3}}{6912}-\frac{1}{6912}\right)^{-n} \left(\left(-3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+9 \,3^{\frac{5}{6}} \left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-48 \,\mathrm{I} \sqrt{3}-48\right)^{-n} \left(\left(\left(-\frac{3^{\frac{5}{6}}}{3}+\mathrm{I} \,3^{\frac{1}{3}}\right) \sqrt{59}-\frac{413 \,\mathrm{I} \,3^{\frac{5}{6}}}{1419}+\frac{413 \,3^{\frac{1}{3}}}{1419}\right) 2^{\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{14160}{473}+\frac{961 \,2^{\frac{1}{3}} \left(\left(\mathrm{I} \,3^{\frac{2}{3}}+3^{\frac{1}{6}}\right) \sqrt{59}-\frac{20001 \,\mathrm{I} \,3^{\frac{1}{6}}}{961}-\frac{6667 \,3^{\frac{2}{3}}}{961}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{2838}\right) \left(-\frac{\left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{12}+\frac{\left(\left(-\mathrm{I} \sqrt{59}-3\right) 18^{\frac{1}{3}}+9 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+3^{\frac{1}{6}} 2^{\frac{1}{3}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{96}\right)^{n}-\frac{961 \left(\left(-3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+9 \,3^{\frac{5}{6}} \left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-48 \,\mathrm{I} \sqrt{3}-48\right)^{-n} \sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}} \left(-\sqrt{59}\, 3^{\frac{1}{6}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+3 \left(81+9 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+8 \,6^{\frac{1}{3}}\right)^{-n} \left(\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-9 \,3^{\frac{5}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+48 \,\mathrm{I} \sqrt{3}-48\right)^{-n} \left(18+2 \sqrt{59}\, \sqrt{3}\right)^{-\frac{n}{3}} \left(-8 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+8 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\left(\left(\mathrm{I} \sqrt{59}-3\right) 18^{\frac{1}{3}}-9 \,\mathrm{I} \,3^{\frac{1}{6}} 2^{\frac{1}{3}}+3^{\frac{1}{6}} 2^{\frac{1}{3}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n} \left(-13824 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-13824 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\left(\left(-1728 \,\mathrm{I} \sqrt{59}-5184\right) 18^{\frac{1}{3}}+15552 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+1728 \,3^{\frac{1}{6}} 2^{\frac{1}{3}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n}}{1419}+\frac{14160 \left(\left(-3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+9 \,3^{\frac{5}{6}} \left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-48 \,\mathrm{I} \sqrt{3}-48\right)^{-n} \left(\frac{1}{\sqrt{59}\, 3^{\frac{1}{6}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-3 \left(81+9 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-8 \,6^{\frac{1}{3}}}\right)^{n} \left(\frac{\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{165888}-\frac{3^{\frac{5}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{18432}+\frac{\mathrm{I} \sqrt{3}}{3456}-\frac{1}{3456}\right)^{-n} \left(\frac{\left(-\mathrm{I} \sqrt{59}+3\right) \left(81+9 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{96}+\frac{3 \,3^{\frac{1}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{32}-\frac{\mathrm{I} \,3^{\frac{5}{6}} 2^{\frac{1}{3}}}{12}+\frac{6^{\frac{1}{3}}}{12}\right)^{n} \left(-8 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-8 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\left(\left(-\mathrm{I} \sqrt{59}-3\right) 18^{\frac{1}{3}}+9 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+3^{\frac{1}{6}} 2^{\frac{1}{3}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n}}{473}+\frac{2 \left(2^{\frac{2}{3}} \left(\sqrt{59}\, 3^{\frac{5}{6}}-\frac{413 \,3^{\frac{1}{3}}}{473}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{6667 \,2^{\frac{1}{3}} 3^{\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{946}\right) \left(-\sqrt{59}\, 3^{\frac{1}{6}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+3 \left(81+9 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+8 \,6^{\frac{1}{3}}\right)^{-n} \left(18+2 \sqrt{59}\, \sqrt{3}\right)^{\frac{n}{3}} \left(\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-9 \,3^{\frac{5}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+48 \,\mathrm{I} \sqrt{3}-48\right)^{-n} \left(\left(\mathrm{I} \sqrt{59}-3\right) \left(81+9 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-9 \,3^{\frac{1}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+8 \,\mathrm{I} \,3^{\frac{5}{6}} 2^{\frac{1}{3}}-8 \,6^{\frac{1}{3}}\right)^{n} \left(\left(-\mathrm{I} \sqrt{59}-3\right) \left(81+9 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+9 \,3^{\frac{1}{6}} \left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-8 \,\mathrm{I} \,3^{\frac{5}{6}} 2^{\frac{1}{3}}-8 \,6^{\frac{1}{3}}\right)^{n} \left(\frac{\left(-3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{1728}+\frac{3^{\frac{5}{6}} \left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{192}-\frac{\mathrm{I} \sqrt{3}}{36}-\frac{1}{36}\right)^{-n}}{3}\right)}{30208} & 0<n \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) = 143
a(7) = 360
a(8) = 871
a(n) = -2*a(n+2)+a(n+3)-1/24*(n+1)*(n^3+5*n^2+18*n+48), 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) = 143
a \! \left(7\right) = 360
a \! \left(8\right) = 871
a \! \left(n \right) = -2 a \! \left(n +2\right)+a \! \left(n +3\right)-\frac{\left(n +1\right) \left(n^{3}+5 n^{2}+18 n +48\right)}{24}, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/17017/
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[17,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[15,x] = F[16,x]*F[4,x]
F[16,x] = F[11,x]+F[14,x]
F[17,x] = F[18,x]+F[2,x]
F[18,x] = F[19,x]+F[20,x]+F[96,x]
F[19,x] = 0
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[38,x]
F[22,x] = F[23,x]+F[7,x]
F[23,x] = F[19,x]+F[24,x]+F[34,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[26,x]+F[30,x]
F[26,x] = F[11,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[11,x]+F[27,x]
F[30,x] = F[14,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[27,x]+F[31,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[36,x]+F[37,x]
F[36,x] = F[11,x]
F[37,x] = F[27,x]
F[38,x] = F[39,x]+F[58,x]
F[39,x] = F[40,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]+F[48,x]
F[42,x] = F[43,x]+F[7,x]
F[43,x] = F[44,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]+F[47,x]
F[46,x] = F[11,x]
F[47,x] = F[14,x]
F[48,x] = F[39,x]+F[49,x]
F[49,x] = F[50,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[52,x]+F[53,x]
F[52,x] = F[43,x]
F[53,x] = F[54,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[56,x]+F[57,x]
F[56,x] = F[43,x]
F[57,x] = F[54,x]
F[58,x] = F[59,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[61,x]+F[67,x]
F[61,x] = F[23,x]+F[62,x]
F[62,x] = F[63,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[65,x]+F[66,x]
F[65,x] = F[27,x]
F[66,x] = F[31,x]
F[67,x] = F[68,x]+F[87,x]
F[68,x] = F[69,x]
F[69,x] = F[4,x]*F[70,x]
F[70,x] = F[71,x]+F[77,x]
F[71,x] = F[23,x]+F[72,x]
F[72,x] = F[73,x]
F[73,x] = F[4,x]*F[74,x]
F[74,x] = F[75,x]+F[76,x]
F[75,x] = F[27,x]
F[76,x] = F[31,x]
F[77,x] = F[68,x]+F[78,x]
F[78,x] = F[79,x]
F[79,x] = F[4,x]*F[80,x]
F[80,x] = F[81,x]+F[82,x]
F[81,x] = F[72,x]
F[82,x] = F[83,x]
F[83,x] = F[4,x]*F[84,x]
F[84,x] = F[85,x]+F[86,x]
F[85,x] = F[72,x]
F[86,x] = F[83,x]
F[87,x] = F[88,x]
F[88,x] = F[4,x]*F[89,x]
F[89,x] = F[90,x]+F[91,x]
F[90,x] = F[62,x]
F[91,x] = F[92,x]
F[92,x] = F[4,x]*F[93,x]
F[93,x] = F[94,x]+F[95,x]
F[94,x] = F[62,x]
F[95,x] = F[92,x]
F[96,x] = F[4,x]*F[97,x]
F[97,x] = F[101,x]+F[98,x]
F[98,x] = F[2,x]+F[99,x]
F[99,x] = F[100,x]
F[100,x] = F[4,x]*F[98,x]
F[101,x] = F[102,x]
F[102,x] = F[103,x]+F[153,x]+F[19,x]
F[103,x] = F[104,x]*F[4,x]
F[104,x] = F[105,x]+F[107,x]
F[105,x] = F[106,x]+F[11,x]
F[106,x] = F[19,x]+F[24,x]+F[28,x]
F[107,x] = F[108,x]+F[123,x]
F[108,x] = F[109,x]
F[109,x] = F[110,x]*F[4,x]
F[110,x] = F[111,x]+F[113,x]
F[111,x] = F[11,x]+F[112,x]
F[112,x] = F[44,x]
F[113,x] = F[108,x]+F[114,x]
F[114,x] = F[115,x]
F[115,x] = F[116,x]*F[4,x]
F[116,x] = F[117,x]+F[118,x]
F[117,x] = F[112,x]
F[118,x] = F[119,x]
F[119,x] = F[120,x]*F[4,x]
F[120,x] = F[121,x]+F[122,x]
F[121,x] = F[112,x]
F[122,x] = F[119,x]
F[123,x] = F[124,x]
F[124,x] = F[125,x]*F[4,x]
F[125,x] = F[126,x]+F[128,x]
F[126,x] = F[106,x]+F[127,x]
F[127,x] = F[63,x]
F[128,x] = F[129,x]+F[144,x]
F[129,x] = F[130,x]
F[130,x] = F[131,x]*F[4,x]
F[131,x] = F[132,x]+F[134,x]
F[132,x] = F[106,x]+F[133,x]
F[133,x] = F[73,x]
F[134,x] = F[129,x]+F[135,x]
F[135,x] = F[136,x]
F[136,x] = F[137,x]*F[4,x]
F[137,x] = F[138,x]+F[139,x]
F[138,x] = F[133,x]
F[139,x] = F[140,x]
F[140,x] = F[141,x]*F[4,x]
F[141,x] = F[142,x]+F[143,x]
F[142,x] = F[133,x]
F[143,x] = F[140,x]
F[144,x] = F[145,x]
F[145,x] = F[146,x]*F[4,x]
F[146,x] = F[147,x]+F[148,x]
F[147,x] = F[127,x]
F[148,x] = F[149,x]
F[149,x] = F[150,x]*F[4,x]
F[150,x] = F[151,x]+F[152,x]
F[151,x] = F[127,x]
F[152,x] = F[149,x]
F[153,x] = F[154,x]*F[4,x]
F[154,x] = F[102,x]+F[2,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_{17}\! \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_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{2}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{96}\! \left(x \right)
F_{19}\! \left(x \right) = 0
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{38}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{7}\! \left(x \right)
F_{23}\! \left(x \right) = F_{19}\! \left(x \right)+F_{24}\! \left(x \right)+F_{34}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{30}\! \left(x \right)
F_{26}\! \left(x \right) = F_{11}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{11}\! \left(x \right)+F_{27}\! \left(x \right)
F_{30}\! \left(x \right) = F_{14}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{27}\! \left(x \right)+F_{31}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{37}\! \left(x \right)
F_{36}\! \left(x \right) = F_{11}\! \left(x \right)
F_{37}\! \left(x \right) = F_{27}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{58}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \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_{48}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{7}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \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_{11}\! \left(x \right)
F_{47}\! \left(x \right) = F_{14}\! \left(x \right)
F_{48}\! \left(x \right) = F_{39}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{53}\! \left(x \right)
F_{52}\! \left(x \right) = F_{43}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{57}\! \left(x \right)
F_{56}\! \left(x \right) = F_{43}\! \left(x \right)
F_{57}\! \left(x \right) = F_{54}\! \left(x \right)
F_{58}\! \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_{67}\! \left(x \right)
F_{61}\! \left(x \right) = F_{23}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \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_{65}\! \left(x \right)+F_{66}\! \left(x \right)
F_{65}\! \left(x \right) = F_{27}\! \left(x \right)
F_{66}\! \left(x \right) = F_{31}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)+F_{87}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{4}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)+F_{77}\! \left(x \right)
F_{71}\! \left(x \right) = F_{23}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{4}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)+F_{76}\! \left(x \right)
F_{75}\! \left(x \right) = F_{27}\! \left(x \right)
F_{76}\! \left(x \right) = F_{31}\! \left(x \right)
F_{77}\! \left(x \right) = F_{68}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{4}\! \left(x \right) F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{81}\! \left(x \right)+F_{82}\! \left(x \right)
F_{81}\! \left(x \right) = F_{72}\! \left(x \right)
F_{82}\! \left(x \right) = F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{4}\! \left(x \right) F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{85}\! \left(x \right)+F_{86}\! \left(x \right)
F_{85}\! \left(x \right) = F_{72}\! \left(x \right)
F_{86}\! \left(x \right) = F_{83}\! \left(x \right)
F_{87}\! \left(x \right) = F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{4}\! \left(x \right) F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = F_{90}\! \left(x \right)+F_{91}\! \left(x \right)
F_{90}\! \left(x \right) = F_{62}\! \left(x \right)
F_{91}\! \left(x \right) = F_{92}\! \left(x \right)
F_{92}\! \left(x \right) = F_{4}\! \left(x \right) F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{94}\! \left(x \right)+F_{95}\! \left(x \right)
F_{94}\! \left(x \right) = F_{62}\! \left(x \right)
F_{95}\! \left(x \right) = F_{92}\! \left(x \right)
F_{96}\! \left(x \right) = F_{4}\! \left(x \right) F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{101}\! \left(x \right)+F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{2}\! \left(x \right)+F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{100}\! \left(x \right)
F_{100}\! \left(x \right) = F_{4}\! \left(x \right) F_{98}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right)
F_{102}\! \left(x \right) = F_{103}\! \left(x \right)+F_{153}\! \left(x \right)+F_{19}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right) F_{4}\! \left(x \right)
F_{104}\! \left(x \right) = F_{105}\! \left(x \right)+F_{107}\! \left(x \right)
F_{105}\! \left(x \right) = F_{106}\! \left(x \right)+F_{11}\! \left(x \right)
F_{106}\! \left(x \right) = F_{19}\! \left(x \right)+F_{24}\! \left(x \right)+F_{28}\! \left(x \right)
F_{107}\! \left(x \right) = F_{108}\! \left(x \right)+F_{123}\! \left(x \right)
F_{108}\! \left(x \right) = F_{109}\! \left(x \right)
F_{109}\! \left(x \right) = F_{110}\! \left(x \right) F_{4}\! \left(x \right)
F_{110}\! \left(x \right) = F_{111}\! \left(x \right)+F_{113}\! \left(x \right)
F_{111}\! \left(x \right) = F_{11}\! \left(x \right)+F_{112}\! \left(x \right)
F_{112}\! \left(x \right) = F_{44}\! \left(x \right)
F_{113}\! \left(x \right) = F_{108}\! \left(x \right)+F_{114}\! \left(x \right)
F_{114}\! \left(x \right) = F_{115}\! \left(x \right)
F_{115}\! \left(x \right) = F_{116}\! \left(x \right) F_{4}\! \left(x \right)
F_{116}\! \left(x \right) = F_{117}\! \left(x \right)+F_{118}\! \left(x \right)
F_{117}\! \left(x \right) = F_{112}\! \left(x \right)
F_{118}\! \left(x \right) = F_{119}\! \left(x \right)
F_{119}\! \left(x \right) = F_{120}\! \left(x \right) F_{4}\! \left(x \right)
F_{120}\! \left(x \right) = F_{121}\! \left(x \right)+F_{122}\! \left(x \right)
F_{121}\! \left(x \right) = F_{112}\! \left(x \right)
F_{122}\! \left(x \right) = F_{119}\! \left(x \right)
F_{123}\! \left(x \right) = F_{124}\! \left(x \right)
F_{124}\! \left(x \right) = F_{125}\! \left(x \right) F_{4}\! \left(x \right)
F_{125}\! \left(x \right) = F_{126}\! \left(x \right)+F_{128}\! \left(x \right)
F_{126}\! \left(x \right) = F_{106}\! \left(x \right)+F_{127}\! \left(x \right)
F_{127}\! \left(x \right) = F_{63}\! \left(x \right)
F_{128}\! \left(x \right) = F_{129}\! \left(x \right)+F_{144}\! \left(x \right)
F_{129}\! \left(x \right) = F_{130}\! \left(x \right)
F_{130}\! \left(x \right) = F_{131}\! \left(x \right) F_{4}\! \left(x \right)
F_{131}\! \left(x \right) = F_{132}\! \left(x \right)+F_{134}\! \left(x \right)
F_{132}\! \left(x \right) = F_{106}\! \left(x \right)+F_{133}\! \left(x \right)
F_{133}\! \left(x \right) = F_{73}\! \left(x \right)
F_{134}\! \left(x \right) = F_{129}\! \left(x \right)+F_{135}\! \left(x \right)
F_{135}\! \left(x \right) = F_{136}\! \left(x \right)
F_{136}\! \left(x \right) = F_{137}\! \left(x \right) F_{4}\! \left(x \right)
F_{137}\! \left(x \right) = F_{138}\! \left(x \right)+F_{139}\! \left(x \right)
F_{138}\! \left(x \right) = F_{133}\! \left(x \right)
F_{139}\! \left(x \right) = F_{140}\! \left(x \right)
F_{140}\! \left(x \right) = F_{141}\! \left(x \right) F_{4}\! \left(x \right)
F_{141}\! \left(x \right) = F_{142}\! \left(x \right)+F_{143}\! \left(x \right)
F_{142}\! \left(x \right) = F_{133}\! \left(x \right)
F_{143}\! \left(x \right) = F_{140}\! \left(x \right)
F_{144}\! \left(x \right) = F_{145}\! \left(x \right)
F_{145}\! \left(x \right) = F_{146}\! \left(x \right) F_{4}\! \left(x \right)
F_{146}\! \left(x \right) = F_{147}\! \left(x \right)+F_{148}\! \left(x \right)
F_{147}\! \left(x \right) = F_{127}\! \left(x \right)
F_{148}\! \left(x \right) = F_{149}\! \left(x \right)
F_{149}\! \left(x \right) = F_{150}\! \left(x \right) F_{4}\! \left(x \right)
F_{150}\! \left(x \right) = F_{151}\! \left(x \right)+F_{152}\! \left(x \right)
F_{151}\! \left(x \right) = F_{127}\! \left(x \right)
F_{152}\! \left(x \right) = F_{149}\! \left(x \right)
F_{153}\! \left(x \right) = F_{154}\! \left(x \right) F_{4}\! \left(x \right)
F_{154}\! \left(x \right) = F_{102}\! \left(x \right)+F_{2}\! \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_17(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))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_11(x) + F_14(x))
Eq(F_17(x), F_18(x) + F_2(x))
Eq(F_18(x), F_19(x) + F_20(x) + F_96(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_38(x))
Eq(F_22(x), F_23(x) + F_7(x))
Eq(F_23(x), F_19(x) + F_24(x) + F_34(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_26(x) + F_30(x))
Eq(F_26(x), F_11(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_11(x) + F_27(x))
Eq(F_30(x), F_14(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_27(x) + F_31(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_36(x) + F_37(x))
Eq(F_36(x), F_11(x))
Eq(F_37(x), F_27(x))
Eq(F_38(x), F_39(x) + F_58(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_48(x))
Eq(F_42(x), F_43(x) + F_7(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_47(x))
Eq(F_46(x), F_11(x))
Eq(F_47(x), F_14(x))
Eq(F_48(x), F_39(x) + F_49(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_53(x))
Eq(F_52(x), F_43(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_57(x))
Eq(F_56(x), F_43(x))
Eq(F_57(x), F_54(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_61(x) + F_67(x))
Eq(F_61(x), F_23(x) + F_62(x))
Eq(F_62(x), F_63(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_65(x) + F_66(x))
Eq(F_65(x), F_27(x))
Eq(F_66(x), F_31(x))
Eq(F_67(x), F_68(x) + F_87(x))
Eq(F_68(x), F_69(x))
Eq(F_69(x), F_4(x)*F_70(x))
Eq(F_70(x), F_71(x) + F_77(x))
Eq(F_71(x), F_23(x) + F_72(x))
Eq(F_72(x), F_73(x))
Eq(F_73(x), F_4(x)*F_74(x))
Eq(F_74(x), F_75(x) + F_76(x))
Eq(F_75(x), F_27(x))
Eq(F_76(x), F_31(x))
Eq(F_77(x), F_68(x) + F_78(x))
Eq(F_78(x), F_79(x))
Eq(F_79(x), F_4(x)*F_80(x))
Eq(F_80(x), F_81(x) + F_82(x))
Eq(F_81(x), F_72(x))
Eq(F_82(x), F_83(x))
Eq(F_83(x), F_4(x)*F_84(x))
Eq(F_84(x), F_85(x) + F_86(x))
Eq(F_85(x), F_72(x))
Eq(F_86(x), F_83(x))
Eq(F_87(x), F_88(x))
Eq(F_88(x), F_4(x)*F_89(x))
Eq(F_89(x), F_90(x) + F_91(x))
Eq(F_90(x), F_62(x))
Eq(F_91(x), F_92(x))
Eq(F_92(x), F_4(x)*F_93(x))
Eq(F_93(x), F_94(x) + F_95(x))
Eq(F_94(x), F_62(x))
Eq(F_95(x), F_92(x))
Eq(F_96(x), F_4(x)*F_97(x))
Eq(F_97(x), F_101(x) + F_98(x))
Eq(F_98(x), F_2(x) + F_99(x))
Eq(F_99(x), F_100(x))
Eq(F_100(x), F_4(x)*F_98(x))
Eq(F_101(x), F_102(x))
Eq(F_102(x), F_103(x) + F_153(x) + F_19(x))
Eq(F_103(x), F_104(x)*F_4(x))
Eq(F_104(x), F_105(x) + F_107(x))
Eq(F_105(x), F_106(x) + F_11(x))
Eq(F_106(x), F_19(x) + F_24(x) + F_28(x))
Eq(F_107(x), F_108(x) + F_123(x))
Eq(F_108(x), F_109(x))
Eq(F_109(x), F_110(x)*F_4(x))
Eq(F_110(x), F_111(x) + F_113(x))
Eq(F_111(x), F_11(x) + F_112(x))
Eq(F_112(x), F_44(x))
Eq(F_113(x), F_108(x) + F_114(x))
Eq(F_114(x), F_115(x))
Eq(F_115(x), F_116(x)*F_4(x))
Eq(F_116(x), F_117(x) + F_118(x))
Eq(F_117(x), F_112(x))
Eq(F_118(x), F_119(x))
Eq(F_119(x), F_120(x)*F_4(x))
Eq(F_120(x), F_121(x) + F_122(x))
Eq(F_121(x), F_112(x))
Eq(F_122(x), F_119(x))
Eq(F_123(x), F_124(x))
Eq(F_124(x), F_125(x)*F_4(x))
Eq(F_125(x), F_126(x) + F_128(x))
Eq(F_126(x), F_106(x) + F_127(x))
Eq(F_127(x), F_63(x))
Eq(F_128(x), F_129(x) + F_144(x))
Eq(F_129(x), F_130(x))
Eq(F_130(x), F_131(x)*F_4(x))
Eq(F_131(x), F_132(x) + F_134(x))
Eq(F_132(x), F_106(x) + F_133(x))
Eq(F_133(x), F_73(x))
Eq(F_134(x), F_129(x) + F_135(x))
Eq(F_135(x), F_136(x))
Eq(F_136(x), F_137(x)*F_4(x))
Eq(F_137(x), F_138(x) + F_139(x))
Eq(F_138(x), F_133(x))
Eq(F_139(x), F_140(x))
Eq(F_140(x), F_141(x)*F_4(x))
Eq(F_141(x), F_142(x) + F_143(x))
Eq(F_142(x), F_133(x))
Eq(F_143(x), F_140(x))
Eq(F_144(x), F_145(x))
Eq(F_145(x), F_146(x)*F_4(x))
Eq(F_146(x), F_147(x) + F_148(x))
Eq(F_147(x), F_127(x))
Eq(F_148(x), F_149(x))
Eq(F_149(x), F_150(x)*F_4(x))
Eq(F_150(x), F_151(x) + F_152(x))
Eq(F_151(x), F_127(x))
Eq(F_152(x), F_149(x))
Eq(F_153(x), F_154(x)*F_4(x))
Eq(F_154(x), F_102(x) + F_2(x))
Pack JSON:
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Specification JSON:
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