0123_0132_1320_3102

Counting sequence:
1, 1, 2, 6, 20, 59, 140, 299, 617, 1244, 2468, 4838, 9397, 18119, 34728, 66230, 125773, 237978, 448862, 844285, 1584205, 2966224, 5543366, 10342196, 19266477, 35843711, 66605034, 123634484, 229276445, 424824766, 786555428, 1455299303, 2690973829, 4973111716, 9186140532, 16960789378, 31303063377, 57752805679, 106517422800, 196401026298, 362038841617, 667215857842, 1229384213734, 2264791548289, 4171540657081, 7682451585696, 14146369403446, 26045817328200, 47949515923433, 88265007158047, 162463330045538, 299011926007712, 550289911996689, 1012669254726910, 1863457779473580, 3428861035907395, 6309005114131441, 11607924647108428, 21356634410200340, 39291373937385246, 72285177660421789, 132981197161706007, 244636552587695144, 450032247122485406, 827862173152268773, 1522879026325813610, 2801344286554171774, 5153021547129399749, 9478779538950169749, 17435661058241884080, 32071585232330628886, 58992782272170461564, 108511098485890637557, 199593600417708339743, 367127183980245592618, 675280526631170384524, 1242080962550471213621, 2284619079915637879870, 4202194043826999970276, 7729243495716281129599, 14216628448006135329005, 26149001697853829519892, 48096404597156223886148, 88464539297962952877306, 162714042409290469944537, 299281686870526969266623, 550471226460858994675056, 1012484873885369270153698, 1862267083440667400438793, 3425271078047138306490370, 6300101325945108150750694, 11587767452244505677284361, 21313348991687471498613537, 39201559530186006885871168, 72103234409991429731425830, 132619055328207690466666256, 243925339840875574972598545, 448650064488342560400181183, 825198436819892993815894530, 1517780336841989652243035960, 2791639446307423242196667593

Generating function in Maple syntax:
-(3*x^14+8*x^13-2*x^12-22*x^11-17*x^10+20*x^9+37*x^8-2*x^7-20*x^6-5*x^5-x^4+4*x^3+4*x^2-4*x+1)/(x^3+x^2+x-1)/(x-1)^2/(x^2+x-1)^2

Generating function in latex syntax:
-\frac{3 x^{14}+8 x^{13}-2 x^{12}-22 x^{11}-17 x^{10}+20 x^{9}+37 x^{8}-2 x^{7}-20 x^{6}-5 x^{5}-x^{4}+4 x^{3}+4 x^{2}-4 x +1}{\left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{2} \left(x^{2}+x -1\right)^{2}}

Generating function in sympy syntax:
(-3*x**14 - 8*x**13 + 2*x**12 + 22*x**11 + 17*x**10 - 20*x**9 - 37*x**8 + 2*x**7 + 20*x**6 + 5*x**5 + x**4 - 4*x**3 - 4*x**2 + 4*x - 1)/((x - 1)**2*(x**2 + x - 1)**2*(x**3 + x**2 + x - 1))

Implicit equation for the generating function in Maple syntax:
(x^3+x^2+x-1)*(x-1)^2*(x^2+x-1)^2*F(x)+3*x^14+8*x^13-2*x^12-22*x^11-17*x^10+20*x^9+37*x^8-2*x^7-20*x^6-5*x^5-x^4+4*x^3+4*x^2-4*x+1 = 0

Implicit equation for the generating function in latex syntax:
\left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{2} \left(x^{2}+x -1\right)^{2} F \! \left(x \right)+3 x^{14}+8 x^{13}-2 x^{12}-22 x^{11}-17 x^{10}+20 x^{9}+37 x^{8}-2 x^{7}-20 x^{6}-5 x^{5}-x^{4}+4 x^{3}+4 x^{2}-4 x +1 = 0

Explicit closed form in Maple syntax:
37/22*Sum(_alpha^(1-n),_alpha = RootOf(_Z^3+_Z^2+_Z-1))+36/11*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^3+_Z^2+_Z-1))+29/11*Sum(_alpha^(-n),_alpha = RootOf(_Z^3+_Z^2+_Z-1))+1/550*(880*n+880)*Sum(_alpha^(-n-2),_alpha = RootOf(_Z^3-2*_Z+1))+1/550*(-990*n-7381)*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^3-2*_Z+1))+367/50*Sum(_alpha^(1-n),_alpha = RootOf(_Z^3-2*_Z+1))+1/550*(-990*n+2189)*Sum(_alpha^(-n),_alpha = RootOf(_Z^3-2*_Z+1))-piecewise(n = 1,-2,n = 3,2,n = 4,5,n = 5,3,0)

Explicit closed form in latex syntax:
\frac{37 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{1-n}\right)}{22}+\frac{36 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{11}+\frac{29 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{11}+\frac{\left(880 n +880\right) \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{3}-2 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -2}\right)}{550}+\frac{\left(-990 n -7381\right) \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{3}-2 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{550}+\frac{367 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{3}-2 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{1-n}\right)}{50}+\frac{\left(-990 n +2189\right) \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{3}-2 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{550}-\left(\left\{\begin{array}{cc}-2 & n =1 \\ 2 & n =3 \\ 5 & n =4 \\ 3 & n =5 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 59
a(6) = 140
a(7) = 299
a(8) = 617
a(9) = 1244
a(10) = 2468
a(11) = 4838
a(12) = 9397
a(13) = 18119
a(14) = 34728
a(n+4) = 1/4*a(n)+3/4*a(n+1)+1/2*a(n+2)-1/2*a(n+3)+3/4*a(n+6)-1/4*a(n+7)+n-7/4, n >= 15

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) = 20
a \! \left(5\right) = 59
a \! \left(6\right) = 140
a \! \left(7\right) = 299
a \! \left(8\right) = 617
a \! \left(9\right) = 1244
a \! \left(10\right) = 2468
a \! \left(11\right) = 4838
a \! \left(12\right) = 9397
a \! \left(13\right) = 18119
a \! \left(14\right) = 34728
a \! \left(n +4\right) = \frac{a \! \left(n \right)}{4}+\frac{3 a \! \left(n +1\right)}{4}+\frac{a \! \left(n +2\right)}{2}-\frac{a \! \left(n +3\right)}{2}+\frac{3 a \! \left(n +6\right)}{4}-\frac{a \! \left(n +7\right)}{4}+n -\frac{7}{4}, \quad n \geq 15

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14872/
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[24,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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[18,x]
F[12,x] = F[13,x]
F[13,x] = F[14,x]*F[4,x]
F[14,x] = F[10,x]+F[15,x]
F[15,x] = F[12,x]+F[16,x]
F[16,x] = F[17,x]
F[17,x] = F[12,x]*F[4,x]
F[18,x] = F[17,x]+F[19,x]+F[20,x]
F[19,x] = 0
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[23,x]
F[22,x] = F[4,x]
F[23,x] = F[18,x]
F[24,x] = F[25,x]+F[36,x]
F[25,x] = F[26,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[28,x]+F[29,x]
F[28,x] = F[1,x]+F[12,x]
F[29,x] = F[25,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]
F[33,x] = F[25,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[25,x]*F[4,x]
F[36,x] = F[19,x]+F[37,x]+F[59,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[39,x]+F[49,x]
F[39,x] = F[40,x]+F[7,x]
F[40,x] = F[19,x]+F[20,x]+F[41,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[11,x]+F[43,x]
F[43,x] = F[44,x]+F[47,x]
F[44,x] = F[45,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[22,x]
F[47,x] = F[48,x]
F[48,x] = F[4,x]*F[44,x]
F[49,x] = F[36,x]+F[50,x]
F[50,x] = F[51,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[53,x]
F[53,x] = F[19,x]+F[35,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[18,x]+F[4,x]
F[57,x] = F[53,x]+F[58,x]
F[58,x] = F[51,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[61,x]+F[62,x]
F[61,x] = F[25,x]+F[53,x]
F[62,x] = F[63,x]+F[67,x]
F[63,x] = F[64,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[66,x]
F[66,x] = F[12,x]+F[44,x]
F[67,x] = 2*F[19,x]+F[68,x]+F[71,x]
F[68,x] = F[4,x]*F[69,x]
F[69,x] = F[70,x]
F[70,x] = F[18,x]+F[47,x]
F[71,x] = F[4,x]*F[63,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_{24}\! \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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{18}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{10}\! \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_{17}\! \left(x \right) = F_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{17}\! \left(x \right)+F_{19}\! \left(x \right)+F_{20}\! \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_{23}\! \left(x \right)
F_{22}\! \left(x \right) = F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{18}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{36}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{29}\! \left(x \right)
F_{28}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{29}\! \left(x \right) = F_{25}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{25}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{19}\! \left(x \right)+F_{37}\! \left(x \right)+F_{59}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{49}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{7}\! \left(x \right)
F_{40}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \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_{11}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{47}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{22}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{49}\! \left(x \right) = F_{36}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \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_{53}\! \left(x \right) = F_{19}\! \left(x \right)+F_{35}\! \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_{18}\! \left(x \right)+F_{4}\! \left(x \right)
F_{57}\! \left(x \right) = F_{53}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{51}\! \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_{62}\! \left(x \right)
F_{61}\! \left(x \right) = F_{25}\! \left(x \right)+F_{53}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)+F_{67}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{12}\! \left(x \right)+F_{44}\! \left(x \right)
F_{67}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{68}\! \left(x \right)+F_{71}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{18}\! \left(x \right)+F_{47}\! \left(x \right)
F_{71}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \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_24(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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_18(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_14(x)*F_4(x))
Eq(F_14(x), F_10(x) + F_15(x))
Eq(F_15(x), F_12(x) + F_16(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_12(x)*F_4(x))
Eq(F_18(x), F_17(x) + F_19(x) + F_20(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_23(x))
Eq(F_22(x), F_4(x))
Eq(F_23(x), F_18(x))
Eq(F_24(x), F_25(x) + F_36(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_28(x) + F_29(x))
Eq(F_28(x), F_1(x) + F_12(x))
Eq(F_29(x), F_25(x) + F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_25(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_25(x)*F_4(x))
Eq(F_36(x), F_19(x) + F_37(x) + F_59(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_39(x) + F_49(x))
Eq(F_39(x), F_40(x) + F_7(x))
Eq(F_40(x), F_19(x) + F_20(x) + F_41(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_11(x) + F_43(x))
Eq(F_43(x), F_44(x) + F_47(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_22(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_4(x)*F_44(x))
Eq(F_49(x), F_36(x) + F_50(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_19(x) + F_35(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_18(x) + F_4(x))
Eq(F_57(x), F_53(x) + F_58(x))
Eq(F_58(x), F_51(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_61(x) + F_62(x))
Eq(F_61(x), F_25(x) + F_53(x))
Eq(F_62(x), F_63(x) + F_67(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_12(x) + F_44(x))
Eq(F_67(x), 2*F_19(x) + F_68(x) + F_71(x))
Eq(F_68(x), F_4(x)*F_69(x))
Eq(F_69(x), F_70(x))
Eq(F_70(x), F_18(x) + F_47(x))
Eq(F_71(x), F_4(x)*F_63(x))
Pack JSON:
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Specification JSON:
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