A351492 - OEIS

%I #26 May 12 2022 15:19:02

%S 1,0,0,3,6,20,180,1134,7980,71280,685440,7165620,82720440,1036404720,

%T 13990472496,202812132600,3141926096400,51795939162240,

%U 905465629762560,16731527824735920,325859956191352800,6671593966263992640,143254214818174152000

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

%H Seiichi Manyama, <a href="/A351492/b351492.txt">Table of n, a(n) for n = 0..450</a>

%F a(0) = 1; a(n) = ((n-1)!/2) * Sum_{k=3..n} k/(k-2) * a(n-k)/(n-k)!.

%F a(n) = n! * Sum_{k=0..floor(n/3)} |Stirling1(n-2*k,k)|/(2^k * (n-2*k)!).

%F a(n) ~ sqrt(2) * n^n / exp(n). - _Vaclav Kotesovec_, May 04 2022

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

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

%o (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!/2*sum(j=3, i, j/(j-2)*v[i-j+1]/(i-j)!)); v;

%o (PARI) a(n) = n!*sum(k=0, n\3, abs(stirling(n-2*k, k, 1))/(2^k*(n-2*k)!));

%Y Cf. A351493, A351505, A353228.

%K nonn

%O 0,4

%A _Seiichi Manyama_, May 02 2022