OFFSET
0,3
FORMULA
a(n) = n! * Sum_{k=0..floor(n/2)} binomial(n,2*k)/k!.
From Vaclav Kotesovec, Mar 17 2023: (Start)
a(n) = (3*n - 2)*a(n-1) - 3*(n-2)*(n-1)*a(n-2) + (n-2)^2*(n-1)*a(n-3).
a(n) ~ 2^(-1/6) * 3^(-1/2) * exp(1/3 - 2^(-1/3)*n^(1/3) + 3*2^(-2/3)*n^(2/3) - n) * n^(n + 1/6) * (1 + 11*2^(1/3)/(27*n^(1/3)) - 79/(3645*2^(1/3)*n^(2/3))). (End)
MATHEMATICA
Table[n! * Sum[Binomial[n, 2*k]/k!, {k, 0, n/2}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 17 2023 *)
With[{nn=20}, CoefficientList[Series[Exp[(x/(1-x))^2]/(1-x), {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Oct 29 2023 *)
PROG
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((x/(1-x))^2)/(1-x)))
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Mar 16 2023
STATUS
approved