A262960 -id:A262960

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A263026

a(n) = Sum_{k=1..n} stirling2(n,k)*((k+1)!)^3/8.

+10
1

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

(list; graph; refs; listen; history; text; internal format)

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) = (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

Cf. A261833, A262960.

KEYWORD

nonn

AUTHOR

Karol A. Penson and Katarzyna Gorska, Oct 08 2015

STATUS

approved

A263158

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

+10
1

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

(list; graph; refs; listen; history; text; internal format)

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.