A380640 Expansion of e.g.f. exp(x*G(2*x)^2) where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.
1, 1, 9, 193, 6673, 319521, 19575001, 1461908449, 128828471073, 13086232224193, 1505486837413801, 193477959856396161, 27472294970916814129, 4271180551913140331233, 721640087945607030774393, 131656978622706616938932641, 25795404137789777215960879681, 5402020596794976601680149234049
Offset: 0
Keywords
Programs
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PARI
a(n) = if(n==0, 1, 2*n!*sum(k=0, n-1, 2^k*binomial(2*n+k, k)/((2*n+k)*(n-k-1)!)));
Formula
a(n) = 2 * n! * Sum_{k=0..n-1} 2^k * binomial(2*n+k,k)/((2*n+k) * (n-k-1)!) for n > 0.
From Vaclav Kotesovec, Jan 29 2025: (Start)
E.g.f. A(x) satisfies x = log(A(x)) * (1 - 2*log(A(x)))^2.
a(n) ~ 3^(3*n - 3/2) * n^(n-1) / (2^(n + 1/2) * exp(n - 1/6)). (End)
a(n) = 2^(n-1)*U(1-n, 2-3*n, 1/2), where U is the Tricomi confluent hypergeometric function. - Stefano Spezia, Jan 29 2025
E.g.f.: exp( Series_Reversion( x*(1-2*x)^2 ) ). - Seiichi Manyama, Mar 16 2025