This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A380640 #20 Mar 16 2025 10:09:03
%S A380640 1,1,9,193,6673,319521,19575001,1461908449,128828471073,
%T A380640 13086232224193,1505486837413801,193477959856396161,
%U A380640 27472294970916814129,4271180551913140331233,721640087945607030774393,131656978622706616938932641,25795404137789777215960879681,5402020596794976601680149234049
%N 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.
%F A380640 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 A380640 From _Vaclav Kotesovec_, Jan 29 2025: (Start)
%F A380640 E.g.f. A(x) satisfies x = log(A(x)) * (1 - 2*log(A(x)))^2.
%F A380640 a(n) ~ 3^(3*n - 3/2) * n^(n-1) / (2^(n + 1/2) * exp(n - 1/6)). (End)
%F A380640 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
%F A380640 E.g.f.: exp( Series_Reversion( x*(1-2*x)^2 ) ). - _Seiichi Manyama_, Mar 16 2025
%o A380640 (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 A380640 Cf. A001764, A380511.
%Y A380640 Cf. A380636, A380639.
%Y A380640 Cf. A080893, A380643.
%K A380640 nonn
%O A380640 0,3
%A A380640 _Seiichi Manyama_, Jan 28 2025