|
| |
|
|
A035051
|
|
Number of labeled rooted connected graphs where every block is a complete graph.
|
|
7
| |
|
|
0, 1, 2, 12, 116, 1555, 26682, 558215, 13781448, 392209380, 12641850510, 455198725025, 18109373455164, 788854833679549, 37343190699472322, 1908871649888004240, 104789417805394595600, 6148562290130009617619
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,3
|
|
|
COMMENTS
| Equivalently, rooted labeled spanning trees in the complete hypergraph on n vertices (all hyperedges having cardinality 2 or greater).
|
|
|
REFERENCES
| Warren D. Smith and David Warme, Paper in preparation, 2002.
|
|
|
LINKS
| T. D. Noe, Table of n, a(n) for n=0..100
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 864
I. M. Gessel and L. H. Kalikow, Hypergraphs and a functional equation...
|
|
|
FORMULA
| Recurrence: a(1) = 1, a(n) = Sum_{k=1}^{n-1} Bell(k) / k! Sum_{a_j > 0, Sum_{j=1}^k a_j = n-1} {{n-1} choose {a_1, a_2, ..., a_k }} \prod_{j=1}^k a(a_j) for n > 1, where Bell(k) = A000110(k). - Warren D. Smith, Feb 23, 1998.
a(n) = sum_{i=0...n-1} S(n-1, i) n^i, where S(N, M) are Stirling numbers of the second kind - David Warme, Mar 25, 1998.
E.g.f. satisfies A(x)=x*exp(exp(A(x))-1).
Let X_{mu} be a Poisson random variable with mean mu: P(X_{mu} = K) = e^{-mu} mu^K / K!. The n-th moment of X_{mu} is E[X_{mu}^n] = sum_{i=0}^n S(n, i) mu^i. Therefore a(n) = E[X_n^{n-1}]. - Langworth Withers, May 25, 2000
|
|
|
CROSSREFS
| Cf. A007549, A007563, A030019, A038052, A038053, A030438.
Sequence in context: A128571 A052696 A107723 * A012628 A012623 A009742
Adjacent sequences: A035048 A035049 A035050 * A035052 A035053 A035054
|
|
|
KEYWORD
| nonn,eigen,nice
|
|
|
AUTHOR
| Christian G. Bower (bowerc(AT)usa.net), Oct 15 1998.
|
| |
|
|
