A035051 Number of labeled rooted connected graphs where every block is a complete graph.
0, 1, 2, 12, 116, 1555, 26682, 558215, 13781448, 392209380, 12641850510, 455198725025, 18109373455164, 788854833679549, 37343190699472322, 1908871649888004240, 104789417805394595600, 6148562290130009617619
Offset: 0
References
- Warren D. Smith and David Warme, Paper in preparation, 2002.
Links
- T. D. Noe, Table of n, a(n) for n=0..100
- Roland Bacher, On the enumeration of labelled hypertrees and of labelled bipartite trees, arXiv:1102.2708 [math.CO], 2011.
- Maryam Bahrani and Jérémie Lumbroso, Enumerations, Forbidden Subgraph Characterizations, and the Split-Decomposition, arXiv:1608.01465, 2016.
- I. M. Gessel and L. H. Kalikow, Hypergraphs and a functional equation...
- INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 864
- D. M. Warme, Spanning Trees in Hypergraphs with Applications to Steiner Trees. PhD thesis, University of Virginia, 1998.
Programs
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Mathematica
f[n_] := Sum[ n^i*StirlingS2[n - 1, i], {i, 0, n - 1}]; Array[f, 18, 0] (* Robert G. Wilson v, Apr 05 2012 *) Table[If[n == 0, 0, BellB[n - 1, n]], {n, 0, 100}] (* Emanuele Munarini, May 23 2014 *) -
Maxima
a(n):=if n=0 then 0 else sum(stirling2(n-1,k)*n^k,k,0,n); makelist(a(n),n,0,12); /* Emanuele Munarini, May 23 2014 */
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PARI
for(n=0,30, print1(sum(k=0,n-1, stirling(n-1,k,2)*n^k), ", ")) \\ G. C. Greubel, Nov 17 2017
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
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
a(n) ~ exp((1/LambertW(1)-2)*n) * n^(n-1) / (sqrt(1+LambertW(1)) * LambertW(1)^(n-1)). - Vaclav Kotesovec, Jan 22 2014
Comments