A354317 - OEIS

%I #16 May 24 2022 08:11:11

%S 1,0,-1,3,-8,20,-34,-126,2514,-28008,285774,-2922810,30858048,

%T -339954264,3920819748,-47319853140,596005041852,-7799132781792,

%U 105344546511684,-1454910026870412,20242465245436128,-276289562032117200,3490199850169557480

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

%F E.g.f.: 1/(1 + x)^(log(1 + x)/2).

%F a(0) = 1; a(n) = -Sum_{k=1..n} binomial(n-1,k-1) * Stirling1(k,2) * a(n-k).

%F a(n) = Sum_{k=0..floor(n/2)} (2*k)! * Stirling1(n,2*k)/((-2)^k * k!).

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

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

%o (PARI) a(n) = sum(k=0, n\2, (2*k)!*stirling(n, 2*k, 1)/((-2)^k*k!));

%Y Cf. A347001, A354318.

%K sign

%O 0,4

%A _Seiichi Manyama_, May 24 2022