|
%I #34 Nov 17 2017 19:39:45
%S 0,1,2,12,116,1555,26682,558215,13781448,392209380,12641850510,
%T 455198725025,18109373455164,788854833679549,37343190699472322,
%U 1908871649888004240,104789417805394595600,6148562290130009617619
%N Number of labeled rooted connected graphs where every block is a complete graph.
%C Equivalently, rooted labeled spanning trees in the complete hypergraph on n vertices (all hyperedges having cardinality 2 or greater).
%D Warren D. Smith and David Warme, Paper in preparation, 2002.
%H T. D. Noe, <a href="/A035051/b035051.txt">Table of n, a(n) for n=0..100</a>
%H R. Bacher, <a href="http://arxiv.org/abs/1102.2708">On the enumeration of labelled hypertrees and of labelled bipartite trees</a>, arXiv:1102.2708v1 [math.CO]
%H Maryam Bahrani and Jérémie Lumbroso, <a href="http://arxiv.org/abs/1608.01465">Enumerations, Forbidden Subgraph Characterizations, and the Split-Decomposition</a>, arXiv:1608.01465, 2016.
%H I. M. Gessel and L. H. Kalikow, <a href="http://arXiv.org/abs/math.CO/0410373">Hypergraphs and a functional equation...</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=864">Encyclopedia of Combinatorial Structures 864</a>
%H D. M. Warme, <a href="http://www.citeulike.org/user/veer/article/1236677">Spanning Trees in Hypergraphs with Applications to Steiner Trees</a>. PhD thesis, University of Virginia, 1998.
%F 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
%F 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
%F E.g.f. satisfies A(x)=x*exp(exp(A(x))-1).
%F 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
%F Dobinski-type formula: a(n) = 1/e^n*sum {k = 0..inf} n^k*k^(n-1)/k!. Cf. A030019 and A052888. For a refinement of this sequence see A210586. - _Peter Bala_, Apr 05 2012
%F a(n) ~ exp((1/LambertW(1)-2)*n) * n^(n-1) / (sqrt(1+LambertW(1)) * LambertW(1)^(n-1)). - _Vaclav Kotesovec_, Jan 22 2014
%t f[n_] := Sum[ n^i*StirlingS2[n - 1, i], {i, 0, n - 1}]; Array[f, 18, 0] (* _Robert G. Wilson v_, Apr 05 2012 *)
%t Table[If[n == 0, 0, BellB[n - 1, n]], {n, 0, 100}] (* _Emanuele Munarini_, May 23 2014 *)
%o (Maxima) a(n):=if n=0 then 0 else sum(stirling2(n-1,k)*n^k,k,0,n);
%o makelist(a(n),n,0,12); /* _Emanuele Munarini_, May 23 2014 */
%o (PARI) for(n=0,30, print1(sum(k=0,n-1, stirling(n-1,k,2)*n^k), ", ")) \\ _G. C. Greubel_, Nov 17 2017
%Y Cf. A007549, A007563, A030019, A038052, A038053, A030438. A052888, A210586.
%K nonn,eigen,nice
%O 0,3
%A _Christian G. Bower_, Oct 15 1998
|
