A351155 Expansion of e.g.f. (1 - x^2/2)^(-x).

1, 0, 0, 3, 0, 15, 90, 210, 2520, 13230, 103950, 873180, 7484400, 72972900, 745404660, 8185126950, 95805309600, 1184852869200, 15538995271800, 214159261516200, 3109622647131000, 47252530639314000, 752635500963746400, 12499951421009052000, 216709136059079664000

Offset

0,4

Links

Table of n, a(n) for n=0..24.

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=2..floor((n+1)/2)} (2*k-1)/((k-1) * 2^(k-1)) * a(n-2*k+1)/(n-2*k+1)!.

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

a(n) ~ sqrt(Pi) * n^(n - 1/2 + sqrt(2)) / (Gamma(sqrt(2)) * exp(n) * 2^(n/2 + sqrt(2) - 1/2)). - Vaclav Kotesovec, May 04 2022

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x^2/2)^(-x)))
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-x*log(1-x^2/2))))
(PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=2, (i+1)\2, (2*j-1)/((j-1)*2^(j-1))*v[i-2*j+2]/(i-2*j+1)!)); v;
(PARI) a(n) = n!*sum(k=0, n\2, abs(stirling(k, n-2*k, 1))/(2^k*k!));

Crossrefs

Cf. A351156, A353226.

Sequence in context: A086479 A091000 A361804 * A013490 A013351 A013407

Adjacent sequences: A351152 A351153 A351154 * A351156 A351157 A351158

Keyword

nonn

Author

Seiichi Manyama, May 02 2022

Status

approved