OFFSET
0,3
FORMULA
a(n) = (n!/(n+1)) * Sum_{k=0..floor(n/3)} (n-3*k)^k * binomial(n+1,n-3*k)/k!.
a(n) ~ 3^(n/3) * (1 + 4*LambertW(3^(1/4)/4))^(n + 3/2) * n^(n-1) / (sqrt(1 + LambertW(3^(1/4)/4)) * 2^(8*n/3 + 4) * exp(n) * LambertW(3^(1/4)/4)^(4*n/3 + 3/2)). - Vaclav Kotesovec, Nov 08 2023
MATHEMATICA
Join[{1}, Table[n!/(n+1) * Sum[(n-3*k)^k * Binomial[n+1, n-3*k]/k!, {k, 0, Floor[n/3]}], {n, 1, 20}]] (* Vaclav Kotesovec, Nov 08 2023 *)
PROG
(PARI) a(n) = n!*sum(k=0, n\3, (n-3*k)^k*binomial(n+1, n-3*k)/k!)/(n+1);
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Aug 31 2023
STATUS
approved