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 A385621 #12 Jul 05 2025 09:58:53
%S A385621 1,1,5,88,5301,1115376,823422553,2109689685664,18480955532693321,
%T A385621 545890697425512822016,53732023859661557312932461,
%U A385621 17451409191554766719804179944960,18549102840350232594174973144785505021,64075761168350693656591055399719635872352256,715103564008507527975398262955305194708318040264449
%N A385621 E.g.f. A(x) satisfies A(x) = exp( x*(A(x) + A(3*x))/2 ).
%F A385621 a(0) = 1; a(n) = (1/2) * Sum_{k=0..n-1} (k+1) * (3^k+1) * binomial(n-1,k) * a(k) * a(n-1-k).
%t A385621 terms = 15; A[_] = 1; Do[A[x_] =Exp[ x*(A[x] + A[3*x])/2 ]+ O[x]^terms // Normal, terms]; CoefficientList[A[x], x]Range[0,terms-1]! (* _Stefano Spezia_, Jul 05 2025 *)
%o A385621 (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, i-1, (j+1)*(3^j+1)*binomial(i-1, j)*v[j+1]*v[i-j])/2); v;
%Y A385621 Cf. A007051, A385622.
%K A385621 nonn
%O A385621 0,3
%A A385621 _Seiichi Manyama_, Jul 05 2025