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%I #11 May 24 2022 08:11:58
%S 1,0,1,3,11,50,273,1687,11505,86004,700445,6163751,58148547,584622766,
%T 6235669629,70286727435,834288853217,10395375065096,135592878107673,
%U 1846897191981835,26212412703559515,386874121137659274,5927186655133112105,94108950154465139807
%N Expansion of e.g.f. exp( x/4 * (exp(2 * x) - 1) ).
%F a(0) = 1; a(n) = Sum_{k=2..n} k * 2^(k-3) * binomial(n-1,k-1) * a(n-k).
%F a(n) = n! * Sum_{k=0..floor(n/2)} 2^(n-3*k) * Stirling2(n-k,k)/(n-k)!.
%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x/4*(exp(2*x)-1))))
%o (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-3)*binomial(i-1, j-1)*v[i-j+1])); v;
%o (PARI) a(n) = n!*sum(k=0, n\2, 2^(n-3*k)*stirling(n-k, k, 2)/(n-k)!);
%Y Cf. A354325.
%K nonn
%O 0,4
%A _Seiichi Manyama_, May 24 2022
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