A354391 - OEIS

%I #10 May 25 2022 09:15:21

%S 1,0,-1,-3,-1,45,269,147,-11341,-101055,-73711,8420247,99423719,

%T 87623445,-13791067291,-202300002453,-202683482821,42194985241545,

%U 738185254885529,805294804942047,-216422419200618961,-4390167368672158755,-5040372451183319251

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

%F a(0) = 1; a(n) = -Sum_{k=1..n} binomial(n,k) * Stirling2(k,2) * a(n-k).

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

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+(exp(x)-1)^2/2)))

%o (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=-sum(j=1, i, binomial(i, j)*stirling(j, 2, 2)*v[i-j+1])); v;

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

%Y Cf. A354392, A354393, A354394.

%Y Cf. A330047, A354389, A354395.

%K sign

%O 0,4

%A _Seiichi Manyama_, May 25 2022