A000558 Generalized Stirling numbers of second kind. (Formerly M4213 N1758)

1, 6, 32, 175, 1012, 6230, 40819, 283944, 2090424, 16235417, 132609666, 1135846062, 10175352709, 95108406130, 925496853980, 9357279554071, 98118527430960, 1065259283215810, 11956366813630835, 138539436100687988, 1655071323662574756, 20361556640795422729

Offset

2,2

Comments

From Olivier Gérard, Mar 25 2009: (Start)

a(n) is the number of hierarchical partitions of a set of n elements into two second level classes : k>1 subsets of [n] are further grouped in two classes.

a(n) is equivalently the number of trees of uniform height 3 with n labeled leaves, and a root of order two. (End)

References

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

T. D. Noe, Table of n, a(n) for n = 2..100

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.

R. Fray, A generating function associated with the generalized Stirling numbers, Fib. Quart. 5 (1967), 356-366.

Formula

E.g.f.: (1/2)*(exp(exp(x)-1)-1)^2. - Vladeta Jovovic, Sep 28 2003

a(n) = Sum_{k=0..n} Stirling2(n,k)*Stirling2(k,2). - Olivier Gérard, Mar 25 2009

a(n) = Sum_{k=1..n-1} binomial(n-1,k) * Bell(k) * Bell(n-k). - Ilya Gutkovskiy, Feb 15 2021

Example

From Olivier Gérard, Mar 25 2009: (Start)
a(2) = 1, since there is only one partition of {1,2} into two classes, and only one way to partition those classes.
a(4) = 32 = 7*1 + 6*3 + 1*7 since there are 7 ways of partitioning {1,2,3,4} into two classes (which cannot be grouped further), 6 ways of partitioning a set of 4 elements into three classes and three ways to partition three classes into two super-classes, etc. (End)

Mathematica

nn = 22; t = Range[0, nn]! CoefficientList[Series[1/2*(Exp[Exp[x] - 1] - 1)^2, {x, 0, nn}], x]; Drop[t, 2] (* T. D. Noe, Aug 10 2012 *)
a[n_] := Sum[StirlingS2[n, k] (2^(k-1)-1), {k, 0, n}];
a /@ Range[2, 100] (* Jean-François Alcover, Mar 30 2021 *)

Crossrefs

Cf. A000110, A000559, A046817.

Cf. A001861 for the related bicolor set partitions. - Olivier Gérard, Mar 25 2009

Sequence in context: A264460 A180037 A277742 * A047763 A259621 A026993

Adjacent sequences: A000555 A000556 A000557 * A000559 A000560 A000561

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from David W. Wilson, Jan 13 2000

Status

approved