OFFSET
0,3
COMMENTS
Generally, for p>=1 is Sum_{k=0..n} (k!)^(2*p) * StirlingS2(n,k)^p asymptotic to c * (n!)^(2*p), where c = 1 + Sum_{n>=1} 1/(Product_{k=1..n} (2*k)^p).
FORMULA
a(n) ~ c * (n!)^4, where c = BesselI(0,1) = 1.266065877752... (see A197036).
MAPLE
a:= n-> add(k!^4*Stirling2(n, k)^2, k=0..n):
seq(a(n), n=0..15); # Alois P. Heinz, Oct 23 2023
MATHEMATICA
Table[Sum[(k!)^4 * StirlingS2[n, k]^2, {k, 0, n}], {n, 0, 20}]
PROG
(PARI) a(n)=sum(k=0, n, k!^4*stirling(n, k, 2)^2) \\ Charles R Greathouse IV, Oct 23 2023
(PARI) a(n)=if(n==0, return(1)); my(Q=x^(n-1), f=1); sum(k=1, n, f*=k; my(t=divrem(Q, x-k)); Q=t[1]; simplify(t[2])^2*f^4) \\ Charles R Greathouse IV, Oct 23 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, May 10 2014
STATUS
approved