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A263026 a(n) = Sum_{k=1..n} Stirling2(n,k)*((k+1)!)^3/8.

%I A263026 #42 May 18 2025 19:03:09
%S A263026 1,28,1810,226558,48859606,16717044358,8536211225830,6206816010688678,
%T A263026 6191950081736354086,8223501207813329312038,
%U A263026 14182148054223247947725350,31102596462109513014876988198,85207893723061275473574262742566,287156553366174285430392015701185318,1174632657911183483067648902342293048870
%N A263026 a(n) = Sum_{k=1..n} Stirling2(n,k)*((k+1)!)^3/8.
%F A263026 Representation as a sum of infinite series of special values of Meijer G functions, a(n) = (1/8)*Sum_{k>=0} MeijerG([[1-k],[]],[[2,2,2],[]],1)*k^n/k!. The Meijer G functions in the above formula cannot be represented through any other special function.
%p A263026 # This program is intended for quick evaluation of a(n)
%p A263026 with(combinat):
%p A263026 a:= n-> add(stirling2(n, k)*((k+1)!)^3, k=1..n)/8:
%p A263026 seq(a(n), n=1..15);
%p A263026 # Maple program for the evaluation and verification of the infinite series representation:
%p A263026 a:= n-> evalf(sum(k^n*evalf(MeijerG([[1-k],[]],[[2,2,2],[]],1))/k!,k=0..infinity)/8); # n=1,2,... .
%p A263026 # This infinite series is slowly converging and the use of 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 = 6000. For example this yields for a(4) = 226558 the approximation 226557.9980714 in about 100 sec. Increasing kmax improves this approximation.
%t A263026 Table[Sum[StirlingS2[n, k] ((k + 1)!)^3/8, {k, n}], {n, 15}] (* _Michael De Vlieger_, Oct 09 2015 *)
%Y A263026 Cf. A261833, A262960.
%K A263026 nonn
%O A263026 1,2
%A A263026 _Karol A. Penson_ and Katarzyna Gorska, Oct 08 2015