OFFSET
1,2
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.
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Karol A. Penson and Katarzyna Gorska, Oct 08 2015
STATUS
approved