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

1, 28, 1810, 226558, 48859606, 16717044358, 8536211225830, 6206816010688678, 6191950081736354086, 8223501207813329312038, 14182148054223247947725350, 31102596462109513014876988198, 85207893723061275473574262742566, 287156553366174285430392015701185318, 1174632657911183483067648902342293048870

Offset: 1

Author

Karol A. Penson and Katarzyna Gorska, Oct 08 2015

Keywords

• nonn

Crossrefs

Cf. A261833, A262960.

Programs

Maple

# This program is intended for quick evaluation of a(n)
with(combinat):
a:= n-> add(stirling2(n, k)*((k+1)!)^3, k=1..n)/8:
seq(a(n), n=1..15);
# Maple program for the evaluation and verification of the infinite series representation:
a:= n-> evalf(sum(k^n*evalf(MeijerG([[1-k],[]],[[2,2,2],[]],1))/k!,k=0..infinity)/8); # n=1,2,... .
# 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.

Mathematica

Table[Sum[StirlingS2[n, k] ((k + 1)!)^3/8, {k, n}], {n, 15}] (* Michael De Vlieger, Oct 09 2015 *)

Formula

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.