0132_1032_1203_1230_3120
Counting sequence:
1, 1, 2, 6, 19, 52, 129, 304, 695, 1558, 3445, 7540, 16371, 35314, 75761, 161776, 344047, 729070, 1540077, 3244012, 6815723, 14286826, 29884393, 62390248, 130023399, 270532582, 562036709, 1166016484, 2415919075, 4999610338, 10334765025, 21340618720, 44023414751, 90731184094, 186831077341, 384399572956, 790273982427, 1623497637850, 3332894621657, 6837587935192, 14018773254103, 28724741275606, 58823872085973, 120396523241428, 246290604621779, 503576325521362, 1029142883598289, 2102266232307664, 4292493394837455, 8760908650119118, 17873661021126605, 36451009484029900, 74309393851613131, 151433537470332874, 308496574474878921, 628252148018184136, 1279022294173220807, 2603080584620146630, 5296233161787703237, 10772610308670226372, 21905508587530092483, 44531593115439464386, 90504338111637487553, 183890979984792092608, 373546567492618420159, 758622350031305310142, 1540303130154747559869, 3126723120493768998844, 6345679961356085755835, 12875827363449267027898, 26120589608372725088185, 52979048979693832241080, 107433837485284428611511, 217819154022362385481654, 441541266148311827480501, 894888448503797767995316, 1813388729421943762059187, 3674001123672583976255410, 7442449577002560856784817, 15073793813319907522117552, 30525376945269386661330863, 61806332527797916556853166, 125123822330114119582089133, 253269959209264812100943788, 512584547516602770075418539, 1037258353229351831897898922, 2098695222850996247289921449, 4245747478486577661568090024, 8588209022542325657112674215, 17369846176222991982178336678, 35126548614722665300262649765, 71026809753998693272337252260, 143601044557104111888298409891, 290296939212421674463844630434, 586783578621270250302184882081, 1185946557635394303353361006496, 2396651916056496212204704497567, 4842821433684407635405373964190, 9784678070511645692802677866397, 19767426547308952229589215608732, 39930993907189226147146150969243
Generating function in Maple syntax:
(2*x^7-4*x^5+x^4-5*x^3+9*x^2-5*x+1)/(2*x-1)^2/(x-1)^2
Generating function in latex syntax:
\frac{2 x^{7}-4 x^{5}+x^{4}-5 x^{3}+9 x^{2}-5 x +1}{\left(2 x -1\right)^{2} \left(x -1\right)^{2}}
Generating function in sympy syntax:
(2*x**7 - 4*x**5 + x**4 - 5*x**3 + 9*x**2 - 5*x + 1)/((x - 1)**2*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
-(2*x-1)^2*(x-1)^2*F(x)+2*x^7-4*x^5+x^4-5*x^3+9*x^2-5*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right)^{2} \left(x -1\right)^{2} F \! \left(x \right)+2 x^{7}-4 x^{5}+x^{4}-5 x^{3}+9 x^{2}-5 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,n = 3,6,1/16*(5*n+4)*2^n-n-1)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 6 & n =3 \\ \frac{\left(5 n +4\right) 2^{n}}{16}-n -1 & \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) = 52
a(6) = 129
a(7) = 304
a(n+2) = -4*a(n)+4*a(n+1)+1-n, 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) = 52
a \! \left(6\right) = 129
a \! \left(7\right) = 304
a \! \left(n +2\right) = -4 a \! \left(n \right)+4 a \! \left(n +1\right)+1-n, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19655/
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[12,x]*F[4,x]
F[4,x] = F[31,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[24,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[1,x]+F[12,x]
F[18,x] = F[14,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[12,x]*F[21,x]
F[21,x] = F[22,x]+F[23,x]
F[22,x] = F[12,x]
F[23,x] = F[19,x]
F[24,x] = F[25,x]+F[26,x]+F[30,x]
F[25,x] = 0
F[26,x] = F[12,x]*F[27,x]
F[27,x] = F[28,x]+F[29,x]
F[28,x] = F[10,x]
F[29,x] = F[24,x]
F[30,x] = F[12,x]*F[13,x]
F[31,x] = F[32,x]+F[45,x]
F[32,x] = F[33,x]
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[35,x]+F[38,x]
F[35,x] = F[1,x]+F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[12,x]*F[35,x]
F[38,x] = F[32,x]+F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[12,x]*F[41,x]
F[41,x] = F[42,x]+F[43,x]
F[42,x] = F[12,x]
F[43,x] = F[44,x]
F[44,x] = F[40,x]
F[45,x] = F[25,x]+F[46,x]+F[59,x]
F[46,x] = F[12,x]*F[47,x]
F[47,x] = F[48,x]+F[52,x]
F[48,x] = F[10,x]+F[49,x]
F[49,x] = F[50,x]
F[50,x] = F[12,x]*F[51,x]
F[51,x] = F[36,x]+F[49,x]
F[52,x] = F[53,x]+F[56,x]
F[53,x] = F[25,x]+F[46,x]+F[54,x]
F[54,x] = F[12,x]*F[55,x]
F[55,x] = F[32,x]+F[53,x]
F[56,x] = F[57,x]
F[57,x] = F[12,x]*F[58,x]
F[58,x] = F[39,x]+F[56,x]
F[59,x] = F[12,x]*F[60,x]
F[60,x] = F[55,x]+F[61,x]
F[61,x] = F[62,x]+F[78,x]
F[62,x] = F[25,x]+F[63,x]+F[69,x]
F[63,x] = F[12,x]*F[64,x]
F[64,x] = F[65,x]+F[68,x]
F[65,x] = F[12,x]+F[66,x]
F[66,x] = F[67,x]
F[67,x] = F[12,x]*F[36,x]
F[68,x] = F[44,x]
F[69,x] = F[12,x]*F[70,x]
F[70,x] = F[71,x]+F[72,x]
F[71,x] = F[32,x]+F[44,x]
F[72,x] = F[62,x]+F[73,x]
F[73,x] = F[74,x]
F[74,x] = F[12,x]*F[75,x]
F[75,x] = F[76,x]+F[77,x]
F[76,x] = F[44,x]
F[77,x] = F[73,x]
F[78,x] = F[25,x]+F[79,x]+F[80,x]+F[84,x]
F[79,x] = 0
F[80,x] = F[12,x]*F[81,x]
F[81,x] = F[82,x]+F[83,x]
F[82,x] = F[53,x]
F[83,x] = F[78,x]
F[84,x] = F[12,x]*F[61,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{31}\! \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_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{24}\! \left(x \right)
F_{14}\! \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_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{12}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \left(x \right)
F_{23}\! \left(x \right) = F_{19}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{26}\! \left(x \right)+F_{30}\! \left(x \right)
F_{25}\! \left(x \right) = 0
F_{26}\! \left(x \right) = F_{12}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{29}\! \left(x \right)
F_{28}\! \left(x \right) = F_{10}\! \left(x \right)
F_{29}\! \left(x \right) = F_{24}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{45}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{38}\! \left(x \right)
F_{35}\! \left(x \right) = F_{1}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{38}\! \left(x \right) = F_{32}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{43}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{40}\! \left(x \right)
F_{45}\! \left(x \right) = F_{25}\! \left(x \right)+F_{46}\! \left(x \right)+F_{59}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{52}\! \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_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{36}\! \left(x \right)+F_{49}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{56}\! \left(x \right)
F_{53}\! \left(x \right) = F_{25}\! \left(x \right)+F_{46}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{32}\! \left(x \right)+F_{53}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{12}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{39}\! \left(x \right)+F_{56}\! \left(x \right)
F_{59}\! \left(x \right) = F_{12}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{55}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{78}\! \left(x \right)
F_{62}\! \left(x \right) = F_{25}\! \left(x \right)+F_{63}\! \left(x \right)+F_{69}\! \left(x \right)
F_{63}\! \left(x \right) = F_{12}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{68}\! \left(x \right)
F_{65}\! \left(x \right) = F_{12}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{12}\! \left(x \right) F_{36}\! \left(x \right)
F_{68}\! \left(x \right) = F_{44}\! \left(x \right)
F_{69}\! \left(x \right) = F_{12}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)+F_{72}\! \left(x \right)
F_{71}\! \left(x \right) = F_{32}\! \left(x \right)+F_{44}\! \left(x \right)
F_{72}\! \left(x \right) = F_{62}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{12}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)+F_{77}\! \left(x \right)
F_{76}\! \left(x \right) = F_{44}\! \left(x \right)
F_{77}\! \left(x \right) = F_{73}\! \left(x \right)
F_{78}\! \left(x \right) = F_{25}\! \left(x \right)+F_{79}\! \left(x \right)+F_{80}\! \left(x \right)+F_{84}\! \left(x \right)
F_{79}\! \left(x \right) = 0
F_{80}\! \left(x \right) = F_{12}\! \left(x \right) F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{82}\! \left(x \right)+F_{83}\! \left(x \right)
F_{82}\! \left(x \right) = F_{53}\! \left(x \right)
F_{83}\! \left(x \right) = F_{78}\! \left(x \right)
F_{84}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_31(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_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_24(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_17(x) + F_18(x))
Eq(F_17(x), F_1(x) + F_12(x))
Eq(F_18(x), F_14(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_12(x)*F_21(x))
Eq(F_21(x), F_22(x) + F_23(x))
Eq(F_22(x), F_12(x))
Eq(F_23(x), F_19(x))
Eq(F_24(x), F_25(x) + F_26(x) + F_30(x))
Eq(F_25(x), 0)
Eq(F_26(x), F_12(x)*F_27(x))
Eq(F_27(x), F_28(x) + F_29(x))
Eq(F_28(x), F_10(x))
Eq(F_29(x), F_24(x))
Eq(F_30(x), F_12(x)*F_13(x))
Eq(F_31(x), F_32(x) + F_45(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_35(x) + F_38(x))
Eq(F_35(x), F_1(x) + F_36(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_12(x)*F_35(x))
Eq(F_38(x), F_32(x) + F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_12(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_43(x))
Eq(F_42(x), F_12(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_40(x))
Eq(F_45(x), F_25(x) + F_46(x) + F_59(x))
Eq(F_46(x), F_12(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_52(x))
Eq(F_48(x), F_10(x) + F_49(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_12(x)*F_51(x))
Eq(F_51(x), F_36(x) + F_49(x))
Eq(F_52(x), F_53(x) + F_56(x))
Eq(F_53(x), F_25(x) + F_46(x) + F_54(x))
Eq(F_54(x), F_12(x)*F_55(x))
Eq(F_55(x), F_32(x) + F_53(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_12(x)*F_58(x))
Eq(F_58(x), F_39(x) + F_56(x))
Eq(F_59(x), F_12(x)*F_60(x))
Eq(F_60(x), F_55(x) + F_61(x))
Eq(F_61(x), F_62(x) + F_78(x))
Eq(F_62(x), F_25(x) + F_63(x) + F_69(x))
Eq(F_63(x), F_12(x)*F_64(x))
Eq(F_64(x), F_65(x) + F_68(x))
Eq(F_65(x), F_12(x) + F_66(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), F_12(x)*F_36(x))
Eq(F_68(x), F_44(x))
Eq(F_69(x), F_12(x)*F_70(x))
Eq(F_70(x), F_71(x) + F_72(x))
Eq(F_71(x), F_32(x) + F_44(x))
Eq(F_72(x), F_62(x) + F_73(x))
Eq(F_73(x), F_74(x))
Eq(F_74(x), F_12(x)*F_75(x))
Eq(F_75(x), F_76(x) + F_77(x))
Eq(F_76(x), F_44(x))
Eq(F_77(x), F_73(x))
Eq(F_78(x), F_25(x) + F_79(x) + F_80(x) + F_84(x))
Eq(F_79(x), 0)
Eq(F_80(x), F_12(x)*F_81(x))
Eq(F_81(x), F_82(x) + F_83(x))
Eq(F_82(x), F_53(x))
Eq(F_83(x), F_78(x))
Eq(F_84(x), F_12(x)*F_61(x))
Pack JSON:
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Specification JSON:
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