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a(n) = Sum_{k=0..n} (k!*StirlingS2(n,k))^3.
4

%I #9 Sep 14 2017 07:45:11

%S 1,1,9,433,63225,18954001,10159366329,8924902306993,11969476975085625,

%T 23232038620328946001,62655369716047066046649,

%U 227268291642918880258797553,1079475019974966974009683584825,6565863403062578428919598754170001

%N a(n) = Sum_{k=0..n} (k!*StirlingS2(n,k))^3.

%C Generally, for p>=1 is Sum_{k=0..n} (k!*StirlingS2(n,k))^p asymptotic to n^(p*n+1/2) * sqrt(Pi/(2*p*(1-log(2))^(p-1))) / (exp(p*n) * log(2)^(p*n+1)).

%H G. C. Greubel, <a href="/A242280/b242280.txt">Table of n, a(n) for n = 0..169</a>

%F a(n) ~ sqrt(Pi/6) * n^(3*n+1/2) / ((1-log(2)) * exp(3*n) * (log(2))^(3*n+1)).

%t Table[Sum[(k!)^3 * StirlingS2[n,k]^3,{k,0,n}],{n,0,20}]

%Y Cf. A000670, A048144.

%K nonn,easy

%O 0,3

%A _Vaclav Kotesovec_, May 10 2014