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
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from David W. Wilson, Jan 13 2000
STATUS
approved