A354311 - OEIS

%I #14 May 24 2022 08:11:39

%S 1,0,2,6,28,160,1056,7784,63568,569664,5542240,58038112,650045760,

%T 7746901760,97790608384,1302349549440,18235836899584,267663541270528,

%U 4107395264113152,65739857693144576,1095095457262013440,18949711553467957248,340036076121127395328

%N Expansion of e.g.f. exp( x/2 * (exp(2 * x) - 1) ).

%F a(0) = 1; a(n) = Sum_{k=2..n} k * 2^(k-2) * binomial(n-1,k-1) * a(n-k).

%F a(n) = n! * Sum_{k=0..floor(n/2)} 2^(n-2*k) * Stirling2(n-k,k)/(n-k)!.

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x/2*(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-2)*binomial(i-1, j-1)*v[i-j+1])); v;

%o (PARI) a(n) = n!*sum(k=0, n\2, 2^(n-2*k)*stirling(n-k, k, 2)/(n-k)!);

%Y Cf. A052506, A354312.

%Y Cf. A351733, A351736, A354309, A354313.

%K sign

%O 0,3

%A _Seiichi Manyama_, May 23 2022