OFFSET
0,4
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
KEYWORD
nonn
AUTHOR
Seiichi Manyama, May 02 2022
STATUS
approved