A242282 a(n) = Sum_{k=0..n} (k!)^4 * StirlingS2(n,k)^2.

1, 1, 17, 1441, 379217, 241351201, 316806826577, 767860003562401, 3168021900014798417, 20904944903800508800801, 210024043938800961464262737, 3086813642229865705833791897761, 64215498113561436496993921529947217, 1839120994194606497461076159930389792801

Offset: 0

Author

Vaclav Kotesovec, May 10 2014

Keywords

• nonn

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).

Crossrefs

Cf. A064618 (p=1), A242283 (p=3).

Cf. A197036.

Programs

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}]

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

Formula

a(n) ~ c * (n!)^4, where c = BesselI(0,1) = 1.266065877752... (see A197036).