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A242282
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a(n) = Sum_{k=0..n} (k!)^4 * StirlingS2(n,k)^2.
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3
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1, 1, 17, 1441, 379217, 241351201, 316806826577, 767860003562401, 3168021900014798417, 20904944903800508800801, 210024043938800961464262737, 3086813642229865705833791897761, 64215498113561436496993921529947217, 1839120994194606497461076159930389792801
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OFFSET
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0,3
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COMMENTS
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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).
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LINKS
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FORMULA
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a(n) ~ c * (n!)^4, where c = BesselI(0,1) = 1.266065877752... (see A197036).
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MAPLE
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a:= n-> add(k!^4*Stirling2(n, k)^2, k=0..n):
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MATHEMATICA
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Table[Sum[(k!)^4 * StirlingS2[n, k]^2, {k, 0, n}], {n, 0, 20}]
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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