A264228 G.f. A(x) satisfies: A(x)^3 = A( x^3/(1-3*x) ), with A(0) = 0.
1, 1, 2, 5, 13, 35, 97, 274, 785, 2275, 6656, 19630, 58295, 174175, 523238, 1579584, 4789919, 14584723, 44577799, 136732988, 420784888, 1298937282, 4021383654, 12483820395, 38853994422, 121220646116, 379062880051, 1187912517953, 3730305167438, 11736596024002, 36994041916973, 116807229667919, 369415244627269, 1170113816365089
Offset: 1
Keywords
Examples
G.f.: A(x) = x + x^2 + 2*x^3 + 5*x^4 + 13*x^5 + 35*x^6 + 97*x^7 + 274*x^8 + 785*x^9 + 2275*x^10 + 6656*x^11 + 19630*x^12 + 58295*x^13 + 174175*x^14 + ... where A(x)^3 = A( x^3/(1-3*x) ). RELATED SERIES. A(x)^3 = x^3 + 3*x^4 + 9*x^5 + 28*x^6 + 87*x^7 + 270*x^8 + 839*x^9 + 2610*x^10 + 8127*x^11 + 25331*x^12 + 79035*x^13 + 246852*x^14 + 771808*x^15 + ... A( x/(1+x+x^2) ) = x + x^4 + 2*x^7 + 6*x^10 + 22*x^13 + 88*x^16 + 367*x^19 + 1570*x^22 + 6843*x^25 + 30271*x^28 + 135530*x^31 + 612852*x^34 + 2794412*x^37 + 12832472*x^40 + ... Let B(x) = x/Series_Reversion(A(x)), then A(x) = x*B(A(x)), where B(x) = 1 + x + x^2 + x^3 - x^5 - x^6 + 2*x^8 + 3*x^9 - 6*x^11 - 9*x^12 + 20*x^14 + 30*x^15 - 71*x^17 - 110*x^18 + 267*x^20 + 419*x^21 - 1041*x^23 + ... Let C0(x) and C2(x) be series trisections of B(x), B(x) = C0(x) + x + C2(x): C0(x) = 1 + x^3 - x^6 + 3*x^9 - 9*x^12 + 30*x^15 - 110*x^18 + 419*x^21 - 1648*x^24 + 6652*x^27 - 27369*x^30 + 114384*x^33 - 484276*x^36 + ... C2(x) = x^2 - x^5 + 2*x^8 - 6*x^11 + 20*x^14 - 71*x^17 + 267*x^20 - 1041*x^23 + 4168*x^26 - 17047*x^29 + 70902*x^32 + ... + (-1)^(n-1)*A370446(n)*x^(3*n-1) + ... then C0(x) = x^2/C2(x).
Links
- Paul D. Hanna, Table of n, a(n) for n = 1..600
Programs
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PARI
{a(n) = my(A=x); for(i=1, n, A = ( subst(A, x, x^3/(1-3*x +x*O(x^n))) )^(1/3) ); polcoeff(A, n)} for(n=1, 40, print1(a(n), ", "))
Formula
G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following.
(1) A(x)^3 = A( x^3/(1-3*x) ).
(2) A( x/(1+3*x) )^3 = A( x^3/(1+3*x)^2 ). - Paul D. Hanna, Mar 25 2023
(3) A( x/(1+x+x^2) )^3 = A( x^3/(1-x^3)^2 ). - Paul D. Hanna, Mar 11 2024
Comments