A262960
a(n) = sum(stirling2(n,k)*(k+2)!*(k+3)!, k=1..n)/144.
Original entry on oeis.org
1, 21, 661, 28941, 1678501, 124467021, 11484880261, 1290503997741, 173495416001701, 27499205820027021, 5075028072491665861, 1078923766195953890541, 261780612944688782844901, 71901410584558939807059021, 22195276604290979611365107461, 7651037112318147566092161607341
Offset: 1
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with(combinat): a:= n-> sum(stirling2(n, k)*(k+2)!*(k+3)!, k=1..n)/144: seq(a(n), n=1..20);
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Table[Sum[StirlingS2[n, k] (k + 2)! (k + 3)!, {k, n}]/144, {n, 16}] (* Michael De Vlieger, Oct 05 2015 *)
A263026
a(n) = Sum_{k=1..n} Stirling2(n,k)*((k+1)!)^3/8.
Original entry on oeis.org
1, 28, 1810, 226558, 48859606, 16717044358, 8536211225830, 6206816010688678, 6191950081736354086, 8223501207813329312038, 14182148054223247947725350, 31102596462109513014876988198, 85207893723061275473574262742566, 287156553366174285430392015701185318, 1174632657911183483067648902342293048870
Offset: 1
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# 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.
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Table[Sum[StirlingS2[n, k] ((k + 1)!)^3/8, {k, n}], {n, 15}] (* Michael De Vlieger, Oct 09 2015 *)
A263158
a(n) = Sum_{k=1..n} Stirling2(n,k)*(k!)^3.
Original entry on oeis.org
1, 9, 241, 15177, 1871761, 400086249, 136109095921, 69234116652297, 50204612238691921, 49984961118827342889, 66285608345755685396401, 114183585213704219683871817, 250186610841184605935378238481, 684906688327788169186039802989929, 2306818395080969813211747978667981681
Offset: 1
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# 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.
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Table[Sum[StirlingS2[n, k] ((k)!)^3, {k, n}], {n, 15}]
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