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%I #15 Oct 23 2023 11:55:07
%S 1,1,17,1441,379217,241351201,316806826577,767860003562401,
%T 3168021900014798417,20904944903800508800801,
%U 210024043938800961464262737,3086813642229865705833791897761,64215498113561436496993921529947217,1839120994194606497461076159930389792801
%N a(n) = Sum_{k=0..n} (k!)^4 * StirlingS2(n,k)^2.
%C 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).
%F a(n) ~ c * (n!)^4, where c = BesselI(0,1) = 1.266065877752... (see A197036).
%p a:= n-> add(k!^4*Stirling2(n,k)^2, k=0..n):
%p seq(a(n), n=0..15); # _Alois P. Heinz_, Oct 23 2023
%t Table[Sum[(k!)^4 * StirlingS2[n,k]^2,{k,0,n}],{n,0,20}]
%o (PARI) a(n)=sum(k=0,n, k!^4*stirling(n,k,2)^2) \\ _Charles R Greathouse IV_, Oct 23 2023
%o (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
%Y Cf. A064618 (p=1), A242283 (p=3).
%Y Cf. A197036.
%K nonn
%O 0,3
%A _Vaclav Kotesovec_, May 10 2014
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