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 A354324 #12 May 25 2024 22:59:49
%S A354324 1,0,1,6,35,220,1623,14294,144393,1605384,19295585,249938062,
%T A354324 3485830299,52134346004,830954821431,14031857352270,249956799370193,
%U A354324 4682845238636560,92038069890608769,1893193762636115990,40659808272769543635,909744112577077608012
%N A354324 Expansion of e.g.f. exp( x/8 * (exp(4 * x) - 1) ).
%F A354324 E.g.f.: exp( x/8 * (exp(4 * x) - 1) ).
%F A354324 a(0) = 1; a(n) = (1/2) * Sum_{k=2..n} k * 4^(k-2) * binomial(n-1,k-1) * a(n-k).
%F A354324 a(n) = n! * Sum_{k=0..floor(n/2)} 4^(n-2*k) * Stirling2(n-k,k)/(2^k * (n-k)!).
%t A354324 Table[n!*Sum[4^(n - 2*k)*StirlingS2[n - k, k]/(2^k*(n - k)!), {k, 0, Floor[n/2]}], {n, 0, 25}] (* _Wesley Ivan Hurt_, May 25 2024 *)
%o A354324 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x/8*(exp(4*x)-1))))
%o A354324 (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j*4^(j-2)*binomial(i-1, j-1)*v[i-j+1])/2); v;
%o A354324 (PARI) a(n) = n!*sum(k=0, n\2, 4^(n-2*k)*stirling(n-k, k, 2)/(2^k*(n-k)!));
%Y A354324 Cf. A354326.
%K A354324 nonn
%O A354324 0,4
%A A354324 _Seiichi Manyama_, May 24 2022