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 A354311 #14 May 24 2022 08:11:39
%S A354311 1,0,2,6,28,160,1056,7784,63568,569664,5542240,58038112,650045760,
%T A354311 7746901760,97790608384,1302349549440,18235836899584,267663541270528,
%U A354311 4107395264113152,65739857693144576,1095095457262013440,18949711553467957248,340036076121127395328
%N A354311 Expansion of e.g.f. exp( x/2 * (exp(2 * x) - 1) ).
%F A354311 a(0) = 1; a(n) = Sum_{k=2..n} k * 2^(k-2) * binomial(n-1,k-1) * a(n-k).
%F A354311 a(n) = n! * Sum_{k=0..floor(n/2)} 2^(n-2*k) * Stirling2(n-k,k)/(n-k)!.
%o A354311 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x/2*(exp(2*x)-1))))
%o A354311 (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j*2^(j-2)*binomial(i-1, j-1)*v[i-j+1])); v;
%o A354311 (PARI) a(n) = n!*sum(k=0, n\2, 2^(n-2*k)*stirling(n-k, k, 2)/(n-k)!);
%Y A354311 Cf. A052506, A354312.
%Y A354311 Cf. A351733, A351736, A354309, A354313.
%K A354311 sign
%O A354311 0,3
%A A354311 _Seiichi Manyama_, May 23 2022