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%I #16 Jan 29 2025 07:57:20
%S 1,1,9,193,6673,319521,19575001,1461908449,128828471073,
%T 13086232224193,1505486837413801,193477959856396161,
%U 27472294970916814129,4271180551913140331233,721640087945607030774393,131656978622706616938932641,25795404137789777215960879681,5402020596794976601680149234049
%N 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.
%F 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.
%F From _Vaclav Kotesovec_, Jan 29 2025: (Start)
%F E.g.f. A(x) satisfies x = log(A(x)) * (1 - 2*log(A(x)))^2.
%F a(n) ~ 3^(3*n - 3/2) * n^(n-1) / (2^(n + 1/2) * exp(n - 1/6)). (End)
%F 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
%o (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)!)));
%Y Cf. A001764, A380511.
%Y Cf. A380636, A380639.
%Y Cf. A080893, A380643.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Jan 28 2025