OFFSET
0,5
FORMULA
a(0) = 1; a(n) = (n-1)! * Sum_{k=2..floor((n+2)/3)} (3*k-2)/(k-1) * a(n-3*k+2)/(n-3*k+2)!.
a(n) = n! * Sum_{k=0..floor(n/3)} |Stirling1(k,n-3*k)|/k!.
a(n) ~ sqrt(2*Pi) * n^(n + 1/2) / (3*exp(n)). - Vaclav Kotesovec, May 04 2022
PROG
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x^3)^(-x)))
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-x*log(1-x^3))))
(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+2)\3, (3*j-2)/(j-1)*v[i-3*j+3]/(i-3*j+2)!)); v;
(PARI) a(n) = n!*sum(k=0, n\3, abs(stirling(k, n-3*k, 1))/k!);
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, May 01 2022
STATUS
approved