A263158 - OEIS

%I #24 Dec 13 2023 08:39:22

%S 1,9,241,15177,1871761,400086249,136109095921,69234116652297,

%T 50204612238691921,49984961118827342889,66285608345755685396401,

%U 114183585213704219683871817,250186610841184605935378238481,684906688327788169186039802989929,2306818395080969813211747978667981681

%N a(n) = Sum_{k=1..n} stirling2(n,k)*(k!)^3.

%F 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 a(n) ~ n!^3. - _Vaclav Kotesovec_, Jul 12 2018

%p # This program is intended for quick evaluation of a(n)

%p with(combinat):

%p a:= n-> add(stirling2(n, k)*((k)!)^3, k=1..n):

%p seq(a(n), n=1..15);

%p # Maple program for the evaluation and verification of the infinite series representation:

%p a:= n-> evalf(sum(k^n*evalf(MeijerG([[1],[]],[[1+k,1+k,1+k],[]],1))/k!, k=0..infinity)); # n=1, 2, ... .

%p # 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 Table[Sum[StirlingS2[n, k] ((k)!)^3, {k, n}], {n, 15}]

%Y Cf. A261833, A262960, A263026.

%Y Cf. A000670, A064618, A316746.

%K nonn

%O 1,2

%A _Karol A. Penson_ and Katarzyna Gorska, Oct 11 2015