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 A354313 #12 May 24 2022 08:11:44
%S A354313 1,0,2,6,40,280,2496,25424,297984,3920256,57349120,922611712,
%T A354313 16193375232,307896882176,6304666798080,138318662000640,
%U A354313 3236895083167744,80483201605795840,2118875812456366080,58882581280649117696,1722441885524719042560
%N A354313 Expansion of e.g.f. 1/(1 - x/2 * (exp(2 * x) - 1)).
%F A354313 a(0) = 1; a(n) = Sum_{k=2..n} k * 2^(k-2) * binomial(n,k) * a(n-k).
%F A354313 a(n) = n! * Sum_{k=0..floor(n/2)} 2^(n-2*k) * k! * Stirling2(n-k,k)/(n-k)!.
%o A354313 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x/2*(exp(2*x)-1))))
%o A354313 (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, j)*v[i-j+1])); v;
%o A354313 (PARI) a(n) = n!*sum(k=0, n\2, 2^(n-2*k)*k!*stirling(n-k, k, 2)/(n-k)!);
%Y A354313 Cf. A052848, A354314.
%Y A354313 Cf. A216794, A353998, A354311.
%K A354313 nonn
%O A354313 0,3
%A A354313 _Seiichi Manyama_, May 23 2022