A263158 a(n) = Sum_{k=1..n} stirling2(n,k)*(k!)^3.

1, 9, 241, 15177, 1871761, 400086249, 136109095921, 69234116652297, 50204612238691921, 49984961118827342889, 66285608345755685396401, 114183585213704219683871817, 250186610841184605935378238481, 684906688327788169186039802989929, 2306818395080969813211747978667981681

Offset

1,2

Links

Table of n, a(n) for n=1..15.

Formula

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.

a(n) ~ n!^3. - Vaclav Kotesovec, Jul 12 2018

Maple

# This program is intended for quick evaluation of a(n)
with(combinat):
a:= n-> add(stirling2(n, k)*((k)!)^3, k=1..n):
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], []], [[1+k, 1+k, 1+k], []], 1))/k!, k=0..infinity)); # n=1, 2, ... .
# 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.

Mathematica

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

Crossrefs

Cf. A261833, A262960, A263026.

Cf. A000670, A064618, A316746.

Sequence in context: A085799 A278858 A274789 * A183903 A251670 A075127

Adjacent sequences: A263155 A263156 A263157 * A263159 A263160 A263161

Keyword

nonn

Author

Karol A. Penson and Katarzyna Gorska, Oct 11 2015

Status

approved