0213_0231_0321_1230_2130

Counting sequence:
1, 1, 2, 6, 19, 56, 156, 418, 1088, 2775, 6976, 17354, 42843, 105178, 257138, 626706, 1523896, 3699053, 8967174, 21716494, 52553003, 127103628, 307277344, 742609394, 1794245448, 4334317795, 10468798964, 25282800498, 61054420195, 147428463854, 355979075910, 859511186914, 2075230571984, 5010393752409, 12096793191850, 29205405794966, 70510226977251, 170230682718880, 410980463238788, 992207925185154, 2395426323391024, 5783115768561647, 13961759382879432, 33706821263062042, 81375745356704651, 196458943675279170, 474294794582513530, 1145050669862065778, 2764400064902463048, 6673858029159819973, 16112129420332508654, 38898141327023889054, 93908457058182572347, 226715138181500776820, 547338885600297206472, 1321393189283122298962, 3190125778984793737208, 7701645694151101895931, 18593418908904668693692, 44888486715294754503362, 108370398231344556207699, 261629294014786231810774, 631629006192904078448654, 1524887343061234190726850, 3681403759744801685432608, 8887694986572893647760561, 21456793961002714094671314, 51801283328141932262519590, 125059361388984370245013043, 301920007525484199916982280, 728899379050587699294133228, 1759718770428365846510143586, 4248336928739034096698907552, 10256392644150489921512496391, 24761122246917490773717820112, 59778637192938718888931081322, 144318396733869708687161385019, 348415430846583640652812118186, 841149258768970521937908235266, 2030713949013438500998890871762, 4902577157952600376740633142616, 11835868267046239455700485217821, 28574313695958346158637137086038, 68984495666160551697495564122414, 166543305041517936549864931632075, 402071105773545798588478431928156, 970685516633395014438832271063792, 2343442139122709168965643120474034, 5657569795030321548372882138547240, 13658581729462019283925681646097683, 32974733254466908671940783452225540, 79608048239338560397740824447096338, 192190829734877968811286820512977987, 463989707712283709689828967557642686, 1120170245165311262974257221775961174, 2704330198053695261435034767508413090, 6528830641292545898093845081423923696, 15761991480675286070452247077533942793, 38052813602710250189874071064699476090, 91867618686219261726155161506949380534, 221788050975375880081844420354000071939

Generating function in Maple syntax:
(x^7-x^6-3*x^5+x^4+2*x^3+4*x^2-4*x+1)/(x^2+2*x-1)/(x^3+x^2+x-1)/(x-1)^2

Generating function in latex syntax:
\frac{x^{7}-x^{6}-3 x^{5}+x^{4}+2 x^{3}+4 x^{2}-4 x +1}{\left(x^{2}+2 x -1\right) \left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{2}}

Generating function in sympy syntax:
(x**7 - x**6 - 3*x**5 + x**4 + 2*x**3 + 4*x**2 - 4*x + 1)/((x - 1)**2*(x**2 + 2*x - 1)*(x**3 + x**2 + x - 1))

Implicit equation for the generating function in Maple syntax:
(x-1)^2*(x^3+x^2+x-1)*(x^2+2*x-1)*F(x)-x^7+x^6+3*x^5-x^4-2*x^3-4*x^2+4*x-1 = 0

Implicit equation for the generating function in latex syntax:
\left(x -1\right)^{2} \left(x^{3}+x^{2}+x -1\right) \left(x^{2}+2 x -1\right) F \! \left(x \right)-x^{7}+x^{6}+3 x^{5}-x^{4}-2 x^{3}-4 x^{2}+4 x -1 = 0

Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/352*(((18*11^(1/2)-66*I)*3^(1/2)+54*I*11^(1/2)-66)*(17+3*11^(1/2)*3^(1/2))^(1/3)-264+((-165*I-27*11^(1/2))*3^(1/2)+81*I*11^(1/2)+165)*(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)+1/352*(((165*I-27*11^(1/2))*3^(1/2)-81*I*11^(1/2)+165)*(17+3*11^(1/2)*3^(1/2))^(2/3)-264+((66*I+18*11^(1/2))*3^(1/2)-54*I*11^(1/2)-66)*(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)+1/352*((54*11^(1/2)*3^(1/2)-330)*(17+3*11^(1/2)*3^(1/2))^(2/3)-36*(17+3*11^(1/2)*3^(1/2))^(1/3)*11^(1/2)*3^(1/2)+132*(17+3*11^(1/2)*3^(1/2))^(1/3)-264)*(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)+1/352*(-198*2^(1/2)+132)*(-1-2^(1/2))^(-n)+1/352*(198*2^(1/2)+132)*(2^(1/2)-1)^(-n)+1/4*n+3/2)

Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\left(\left(18 \sqrt{11}-66 \,\mathrm{I}\right) \sqrt{3}+54 \,\mathrm{I} \sqrt{11}-66\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}-264+\left(\left(-165 \,\mathrm{I}-27 \sqrt{11}\right) \sqrt{3}+81 \,\mathrm{I} \sqrt{11}+165\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}}{352}\\+\\\frac{\left(\left(\left(165 \,\mathrm{I}-27 \sqrt{11}\right) \sqrt{3}-81 \,\mathrm{I} \sqrt{11}+165\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}-264+\left(\left(66 \,\mathrm{I}+18 \sqrt{11}\right) \sqrt{3}-54 \,\mathrm{I} \sqrt{11}-66\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}}{352}\\+\\\frac{\left(\left(54 \sqrt{11}\, \sqrt{3}-330\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}-36 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{11}\, \sqrt{3}+132 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}-264\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}}{352}\\+\frac{\left(-198 \sqrt{2}+132\right) \left(-1-\sqrt{2}\right)^{-n}}{352}+\frac{\left(198 \sqrt{2}+132\right) \left(\sqrt{2}-1\right)^{-n}}{352}+\frac{n}{4}\\+\frac{3}{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) = 56
a(6) = 156
a(7) = 418
a(n+2) = -1/2*a(n)-3/2*a(n+1)+3/2*a(n+4)-1/2*a(n+5)+1/2*n+3, 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) = 156
a \! \left(7\right) = 418
a \! \left(n +2\right) = -\frac{a \! \left(n \right)}{2}-\frac{3 a \! \left(n +1\right)}{2}+\frac{3 a \! \left(n +4\right)}{2}-\frac{a \! \left(n +5\right)}{2}+\frac{n}{2}+3, \quad n \geq 8

Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/20258/
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[10,x]*F[4,x]
F[4,x] = F[14,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[10,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]+F[11,x]
F[10,x] = x
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[13,x]
F[13,x] = F[9,x]
F[14,x] = F[15,x]+F[28,x]
F[15,x] = F[16,x]
F[16,x] = F[10,x]*F[17,x]
F[17,x] = F[18,x]+F[22,x]
F[18,x] = F[1,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[10,x]*F[21,x]
F[21,x] = F[1,x]+F[10,x]
F[22,x] = F[15,x]+F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[10,x]*F[25,x]
F[25,x] = F[15,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[10,x]*F[15,x]
F[28,x] = F[29,x]+F[30,x]+F[44,x]
F[29,x] = 0
F[30,x] = F[10,x]*F[31,x]
F[31,x] = F[32,x]+F[36,x]
F[32,x] = F[33,x]+F[6,x]
F[33,x] = F[12,x]+F[29,x]+F[34,x]
F[34,x] = F[10,x]*F[35,x]
F[35,x] = F[11,x]+F[6,x]
F[36,x] = F[28,x]+F[37,x]
F[37,x] = 2*F[29,x]+F[38,x]+F[41,x]
F[38,x] = F[10,x]*F[39,x]
F[39,x] = F[28,x]+F[40,x]
F[40,x] = F[41,x]
F[41,x] = F[10,x]*F[42,x]
F[42,x] = F[43,x]
F[43,x] = F[26,x]+F[40,x]
F[44,x] = F[10,x]*F[45,x]
F[45,x] = F[14,x]+F[46,x]
F[46,x] = F[47,x]+F[56,x]
F[47,x] = F[27,x]+F[29,x]+F[48,x]
F[48,x] = F[10,x]*F[49,x]
F[49,x] = F[50,x]+F[53,x]
F[50,x] = F[10,x]+F[51,x]
F[51,x] = F[52,x]
F[52,x] = x^2
F[53,x] = F[26,x]+F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[10,x]*F[26,x]
F[56,x] = 2*F[29,x]+F[57,x]+F[61,x]
F[57,x] = F[10,x]*F[58,x]
F[58,x] = F[59,x]+F[60,x]
F[59,x] = F[11,x]
F[60,x] = F[40,x]
F[61,x] = F[10,x]*F[62,x]
F[62,x] = F[46,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_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{14}\! \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_{10}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{5}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{11}\! \left(x \right)
F_{10}\! \left(x \right) = x
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{9}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{28}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{10}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{22}\! \left(x \right)
F_{18}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{10}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{22}\! \left(x \right) = F_{15}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{10}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{15}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{10}\! \left(x \right) F_{15}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{30}\! \left(x \right)+F_{44}\! \left(x \right)
F_{29}\! \left(x \right) = 0
F_{30}\! \left(x \right) = F_{10}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{36}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{6}\! \left(x \right)
F_{33}\! \left(x \right) = F_{12}\! \left(x \right)+F_{29}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{10}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{11}\! \left(x \right)+F_{6}\! \left(x \right)
F_{36}\! \left(x \right) = F_{28}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = 2 F_{29}\! \left(x \right)+F_{38}\! \left(x \right)+F_{41}\! \left(x \right)
F_{38}\! \left(x \right) = F_{10}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{28}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{10}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{26}\! \left(x \right)+F_{40}\! \left(x \right)
F_{44}\! \left(x \right) = F_{10}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{14}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{56}\! \left(x \right)
F_{47}\! \left(x \right) = F_{27}\! \left(x \right)+F_{29}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{10}\! \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_{10}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = x^{2}
F_{53}\! \left(x \right) = F_{26}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{10}\! \left(x \right) F_{26}\! \left(x \right)
F_{56}\! \left(x \right) = 2 F_{29}\! \left(x \right)+F_{57}\! \left(x \right)+F_{61}\! \left(x \right)
F_{57}\! \left(x \right) = F_{10}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{60}\! \left(x \right)
F_{59}\! \left(x \right) = F_{11}\! \left(x \right)
F_{60}\! \left(x \right) = F_{40}\! \left(x \right)
F_{61}\! \left(x \right) = F_{10}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{46}\! \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_10(x)*F_4(x))
Eq(F_4(x), F_14(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_10(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_11(x))
Eq(F_10(x), x)
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_13(x))
Eq(F_13(x), F_9(x))
Eq(F_14(x), F_15(x) + F_28(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_10(x)*F_17(x))
Eq(F_17(x), F_18(x) + F_22(x))
Eq(F_18(x), F_1(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_10(x)*F_21(x))
Eq(F_21(x), F_1(x) + F_10(x))
Eq(F_22(x), F_15(x) + F_23(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_10(x)*F_25(x))
Eq(F_25(x), F_15(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_10(x)*F_15(x))
Eq(F_28(x), F_29(x) + F_30(x) + F_44(x))
Eq(F_29(x), 0)
Eq(F_30(x), F_10(x)*F_31(x))
Eq(F_31(x), F_32(x) + F_36(x))
Eq(F_32(x), F_33(x) + F_6(x))
Eq(F_33(x), F_12(x) + F_29(x) + F_34(x))
Eq(F_34(x), F_10(x)*F_35(x))
Eq(F_35(x), F_11(x) + F_6(x))
Eq(F_36(x), F_28(x) + F_37(x))
Eq(F_37(x), 2*F_29(x) + F_38(x) + F_41(x))
Eq(F_38(x), F_10(x)*F_39(x))
Eq(F_39(x), F_28(x) + F_40(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_10(x)*F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_26(x) + F_40(x))
Eq(F_44(x), F_10(x)*F_45(x))
Eq(F_45(x), F_14(x) + F_46(x))
Eq(F_46(x), F_47(x) + F_56(x))
Eq(F_47(x), F_27(x) + F_29(x) + F_48(x))
Eq(F_48(x), F_10(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_53(x))
Eq(F_50(x), F_10(x) + F_51(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), x**2)
Eq(F_53(x), F_26(x) + F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_10(x)*F_26(x))
Eq(F_56(x), 2*F_29(x) + F_57(x) + F_61(x))
Eq(F_57(x), F_10(x)*F_58(x))
Eq(F_58(x), F_59(x) + F_60(x))
Eq(F_59(x), F_11(x))
Eq(F_60(x), F_40(x))
Eq(F_61(x), F_10(x)*F_62(x))
Eq(F_62(x), F_46(x))
Pack JSON:
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Specification JSON:
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