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A263158 a(n) = Sum_{k=1..n} Stirling2(n,k)*(k!)^3.

%I A263158 #27 May 18 2025 19:02:58
%S A263158 1,9,241,15177,1871761,400086249,136109095921,69234116652297,
%T A263158 50204612238691921,49984961118827342889,66285608345755685396401,
%U A263158 114183585213704219683871817,250186610841184605935378238481,684906688327788169186039802989929,2306818395080969813211747978667981681
%N A263158 a(n) = Sum_{k=1..n} Stirling2(n,k)*(k!)^3.
%F A263158 Representation as a sum of infinite series of special values of Meijer G functions, a(n) = Sum_{k>=0} MeijerG([[1],[]],[[1+k,1+k,1+k],[]],1)*k^n/k!. The Meijer G functions in the above formula cannot be represented through any other special function.
%F A263158 a(n) ~ n!^3. - _Vaclav Kotesovec_, Jul 12 2018
%p A263158 # This program is intended for quick evaluation of a(n)
%p A263158 with(combinat):
%p A263158 a:= n-> add(stirling2(n, k)*((k)!)^3, k=1..n):
%p A263158 seq(a(n), n=1..15);
%p A263158 # Maple program for the evaluation and verification of the infinite series representation:
%p A263158 a:= n-> evalf(sum(k^n*evalf(MeijerG([[1],[]],[[1+k,1+k,1+k],[]],1))/k!, k=0..infinity)); # n=1, 2, ... .
%p A263158 # This infinite series is slowly converging and the use of the above formula will presumably not give the result in a reasonable time. Instead it is practical to replace the upper summation limit k = infinity by some kmax, say kmax = 5000. For example, this yields for a(3) = 241 the approximation 240.99999999948 in about 90 sec. Increasing kmax improves this approximation.
%t A263158 Table[Sum[StirlingS2[n, k] ((k)!)^3, {k, n}], {n, 15}]
%Y A263158 Cf. A261833, A262960, A263026.
%Y A263158 Cf. A000670, A064618, A316746.
%K A263158 nonn
%O A263158 1,2
%A A263158 _Karol A. Penson_ and Katarzyna Gorska, Oct 11 2015