0132_1230_1302_2103
Counting sequence:
1, 1, 2, 6, 20, 59, 156, 397, 989, 2422, 5858, 14038, 33407, 79084, 186465, 438293, 1027764, 2405552, 5622194, 13125129, 30613594, 71354423, 166221847, 387049676, 900941488, 2096568716, 4877855725, 11346857598, 26391521063, 61377263967, 142729744994, 331889016986, 771700311684, 1794265059319, 4171674260444, 9698911563241, 22548972919133, 52423200860074, 121875039553754, 283335913768498, 658695989183583, 1531319000849840, 3559952109266961, 8276008434008753, 19239613195664588, 44727093229605708, 103978647362905570, 241722445969263653, 561939117588450658, 1306354752142098011, 3036915169491983047, 7059987449182763360, 16412509434334676032, 38154510277434624720, 88698576117152239373, 206199358312189492962, 479355692468605067327, 1114367414173097200067, 2590591093738953493866, 6022395759947368871254, 14000375476264585828556, 32546931464818042497307, 75662449248387780691268, 175893878770214657155693, 408903703265804792075285, 950585864017429452332326, 2209844179268491193470074, 5137264724513192180508854, 11942692067986450692801303, 27763391649622256472894812, 64542056853271057574199785, 150042081748806332478141237, 348805527641035155690096796, 810874484466415095988524848, 1885054494925419784303547386, 4382220071183090045095209449, 10187425759110561102678920962, 23682891723594702645873299303, 55056043896094196589564056095, 127989774317201361475566551772, 297540127466059910245317102152, 691696878732266457434403370524, 1608000157554323298595977891413, 3738146846292384045884172838958, 8690137109173496790982551452079, 20202117804763099912890436153231, 46964226064808356483512842129114, 109178579735591811822865404198586, 253809404958287988435623776474508, 590035281608947975853863837539463, 1371665614904204651650707848901332, 3188735686928176324547891419500697, 7412911113459042145910703129057381, 17232927583032468162006985384325066, 40061696212064131073049844803833730, 93132144589024597816783201979452514, 216505968935980222518016191399284807, 503315313860352559375330643424069456, 1170066148331959109116037602726191337, 2720073786273388888758320484068375969, 6323404376131071699652163143169857156
Generating function in Maple syntax:
-(2*x^8+2*x^7-4*x^6-3*x^5+4*x^4-x^3+5*x^2-4*x+1)/(x-1)/(x^3-2*x^2+3*x-1)/(x^3+x^2+x-1)
Generating function in latex syntax:
-\frac{2 x^{8}+2 x^{7}-4 x^{6}-3 x^{5}+4 x^{4}-x^{3}+5 x^{2}-4 x +1}{\left(x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right) \left(x^{3}+x^{2}+x -1\right)}
Generating function in sympy syntax:
(-2*x**8 - 2*x**7 + 4*x**6 + 3*x**5 - 4*x**4 + x**3 - 5*x**2 + 4*x - 1)/((x - 1)*(x**3 - 2*x**2 + 3*x - 1)*(x**3 + x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(x^3-2*x^2+3*x-1)*(x^3+x^2+x-1)*F(x)+2*x^8+2*x^7-4*x^6-3*x^5+4*x^4-x^3+5*x^2-4*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right) \left(x^{3}+x^{2}+x -1\right) F \! \left(x \right)+2 x^{8}+2 x^{7}-4 x^{6}-3 x^{5}+4 x^{4}-x^{3}+5 x^{2}-4 x +1 = 0
Explicit closed form in Maple syntax:
37121/5566*Sum(_alpha^(-n+5),_alpha = RootOf(_Z^7-2*_Z^6+3*_Z^5-3*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1))-55035/5566*Sum(_alpha^(-n+4),_alpha = RootOf(_Z^7-2*_Z^6+3*_Z^5-3*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1))+40279/2783*Sum(_alpha^(-n+3),_alpha = RootOf(_Z^7-2*_Z^6+3*_Z^5-3*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1))-66905/5566*Sum(_alpha^(-n+2),_alpha = RootOf(_Z^7-2*_Z^6+3*_Z^5-3*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1))+6863/253*Sum(_alpha^(1-n),_alpha = RootOf(_Z^7-2*_Z^6+3*_Z^5-3*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1))-105166/2783*Sum(_alpha^(-n),_alpha = RootOf(_Z^7-2*_Z^6+3*_Z^5-3*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1))+69173/5566*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^7-2*_Z^6+3*_Z^5-3*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1))-2*piecewise(n = 0,3,n = 1,1,0)
Explicit closed form in latex syntax:
\frac{37121 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +5}\right)}{5566}-\frac{55035 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +4}\right)}{5566}+\frac{40279 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +3}\right)}{2783}-\frac{66905 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{5566}+\frac{6863 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{1-n}\right)}{253}-\frac{105166 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{2783}+\frac{69173 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{5566}-2 \left(\left\{\begin{array}{cc}3 & n =0 \\ 1 & n =1 \\ 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) = 156
a(7) = 397
a(8) = 989
a(n+6) = -a(n)+a(n+1)-2*a(n+2)+a(n+3)-4*a(n+4)+4*a(n+5)+2, 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) = 20
a \! \left(5\right) = 59
a \! \left(6\right) = 156
a \! \left(7\right) = 397
a \! \left(8\right) = 989
a \! \left(n +6\right) = -a \! \left(n \right)+a \! \left(n +1\right)-2 a \! \left(n +2\right)+a \! \left(n +3\right)-4 a \! \left(n +4\right)+4 a \! \left(n +5\right)+2, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15610/
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[29,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[16,x]+F[24,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[18,x]+F[21,x]
F[18,x] = F[11,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[18,x]*F[4,x]
F[21,x] = F[22,x]
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[7,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[26,x]+F[7,x]
F[26,x] = F[27,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[18,x]
F[29,x] = F[2,x]+F[30,x]
F[30,x] = F[15,x]+F[31,x]+F[56,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]+F[41,x]
F[33,x] = F[34,x]+F[7,x]
F[34,x] = F[15,x]+F[16,x]+F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[25,x]+F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[23,x]+F[40,x]
F[40,x] = F[38,x]
F[41,x] = F[42,x]+F[47,x]
F[42,x] = F[43,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[45,x]+F[46,x]
F[45,x] = F[2,x]
F[46,x] = F[42,x]
F[47,x] = F[48,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[50,x]+F[51,x]
F[50,x] = F[42,x]
F[51,x] = F[52,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[54,x]+F[55,x]
F[54,x] = F[42,x]
F[55,x] = F[52,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[58,x]+F[63,x]
F[58,x] = F[2,x]+F[59,x]
F[59,x] = F[60,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[62,x]
F[62,x] = F[11,x]+F[26,x]
F[63,x] = F[42,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_{29}\! \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_{16}\! \left(x \right)+F_{24}\! \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_{18}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = F_{11}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{7}\! \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_{7}\! \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_{18}\! \left(x \right)
F_{29}\! \left(x \right) = F_{2}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{15}\! \left(x \right)+F_{31}\! \left(x \right)+F_{56}\! \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_{41}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{7}\! \left(x \right)
F_{34}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{25}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{23}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{38}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{47}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{46}\! \left(x \right)
F_{45}\! \left(x \right) = F_{2}\! \left(x \right)
F_{46}\! \left(x \right) = F_{42}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{51}\! \left(x \right)
F_{50}\! \left(x \right) = F_{42}\! \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_{55}\! \left(x \right)
F_{54}\! \left(x \right) = F_{42}\! \left(x \right)
F_{55}\! \left(x \right) = F_{52}\! \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_{63}\! \left(x \right)
F_{58}\! \left(x \right) = F_{2}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{11}\! \left(x \right)+F_{26}\! \left(x \right)
F_{63}\! \left(x \right) = F_{42}\! \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_29(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) + F_16(x) + F_24(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_21(x))
Eq(F_18(x), F_11(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_18(x)*F_4(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_7(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_26(x) + F_7(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_18(x))
Eq(F_29(x), F_2(x) + F_30(x))
Eq(F_30(x), F_15(x) + F_31(x) + F_56(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_33(x) + F_41(x))
Eq(F_33(x), F_34(x) + F_7(x))
Eq(F_34(x), F_15(x) + F_16(x) + F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_25(x) + F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_23(x) + F_40(x))
Eq(F_40(x), F_38(x))
Eq(F_41(x), F_42(x) + F_47(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_46(x))
Eq(F_45(x), F_2(x))
Eq(F_46(x), F_42(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_51(x))
Eq(F_50(x), F_42(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) + F_55(x))
Eq(F_54(x), F_42(x))
Eq(F_55(x), F_52(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_58(x) + F_63(x))
Eq(F_58(x), F_2(x) + F_59(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_11(x) + F_26(x))
Eq(F_63(x), F_42(x))
Pack JSON:
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Specification JSON:
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