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A370534
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Expansion of g.f. A(x) = G( x*(1 + 3*x)*G(x)^2 )^(1/3) = G( x^2*(1 + 5*x)*G(x)^3 )^(1/5), where G(x) is the g.f. of A370533.
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4
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1, 1, -11, 45, -135, 297, -198, -1872, 15705, -103695, 556749, -2275131, 6904116, -15392760, 38895675, -278647283, 2238863647, -12869710817, 53512462775, -155088202345, 174598433696, 1545643834686, -15193843495341, 84540206058435, -303767647556085, 263679736843551
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OFFSET
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1,3
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LINKS
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EXAMPLE
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G.f.: A(x) = x + x^2 - 11*x^3 + 45*x^4 - 135*x^5 + 297*x^6 - 198*x^7 - 1872*x^8 + 15705*x^9 - 103695*x^10 + 556749*x^11 - 2275131*x^12 + ...
where A(x) = G( x*(1 + 3*x)*G(x)^2 )^(1/3) = G( x^2*(1 + 5*x)*G(x)^3 )^(1/5)
and G(x) is the g.f. of A370533, which begins
G(x) = x - 15*x^3 + 80*x^4 - 255*x^5 + 432*x^6 + 1020*x^7 - 12510*x^8 + 71595*x^9 - 354070*x^10 + 1570104*x^11 - 5622420*x^12 + ...
RELATED SERIES.
A(x)^3 = G( x*(1 + 3*x)*G(x)^2 ) = x^3 + 3*x^4 - 30*x^5 + 70*x^6 + 195*x^7 - 2391*x^8 + 11467*x^9 - 30645*x^10 + 11340*x^11 + ...
A(x)^5 = G( x^2*(1 + 5*x)*G(x)^3 ) = x^5 + 5*x^6 - 45*x^7 + 15*x^8 + 1110*x^9 - 6354*x^10 + 12315*x^11 + 64365*x^12 + ...
A(x)^15 = x^15 + 15*x^16 - 60*x^17 - 1180*x^18 + 6480*x^19 + 41688*x^20 - 480825*x^21 + 497475*x^22 + 16467975*x^23 + ...
where A(x)^15 = G( x*(1 + 3*x)*G(x)^2 )^5 = G( x^2*(1 + 5*x)*G(x)^3 )^3.
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PROG
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(PARI) {a(n) = my(A, G, V=[1]); for(i=1, n+1, V = concat(V, 0); G = x*Ser(V);
V[#V] = polcoeff( subst(G, x, x^2*(1 + 5*x)*G^3 )^3 - subst(G, x, x*(1 + 3*x)*G^2 )^5, #V+14); );
A = subst(G, x, x*(1 + 3*x)*G^2 )^(1/3); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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