Search: id:A020556
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%I A020556
%S A020556 1,1,7,87,1657,43833,1515903,65766991,3473600465,218310229201,
%T A020556 16035686850327,1356791248984295,130660110400259849,
%U A020556 14177605780945123273,1718558016836289502159,230999008481288064430879
%N A020556 Number of oriented multigraphs on n labeled arcs (without loops).
%C A020556 Generalized Bell numbers: a(n)=sum(A078739(n,k),k=2..2*n),n>=1.
%D A020556 G. Labelle, Counting enriched multigraphs..., Discrete Math., 217 (2000), 
               237-248.
%D A020556 P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal 
               ordering problem, Phys. Lett. A 309 (2003) 198-205.
%D A020556 G. Paquin, D\'enombrement de multigraphes enrichis, M\'emoire, Math. 
               Dept., Univ. Qu\'ebec \`a Montr\'eal, 2004.
%D A020556 M. Schork, On the combinatorics of normal ordering bosonic operators 
               and deforming it, J. Phys. A 36 (2003) 4651-4665.
%H A020556 P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://arXiv.org/
               abs/quant-ph/0212072">The Boson Normal Ordering Problem and Generalized 
               Bell Numbers</a>
%H A020556 P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://www.arXiv.org/
               abs/quant-ph/0402027">The general boson normal ordering problem.</
               a>
%F A020556 Sum( (k+2)!^n /(k+2)!*(k!^n)*exp(1)), k = 0 .. infinity, n>=1.
%F A020556 (sum(((k*(k-1))^n)/k!, k=2..infinity)/exp(1), n>=1. a(0) := 1. (from 
               eq.(26) with r=2 of the Schork reference.)
%F A020556 E.g.f.: (sum((exp(k*(k-1)*x))/k!, k=2..infinity)+2)/exp(1) (from top 
               of p. 4656 of the Schork reference).
%F A020556 a(n) = Sum_{k=0..n} (-1)^k*binomial(n, k)*Bell(2*n-k). - Vladeta Jovovic 
               (vladeta(AT)eunet.rs), May 02 2004
%t A020556 f[n_] := f[n] = Sum[(k + 2)!^n/((k + 2)!*(k!^n)*E), {k, 0, Infinity}]; 
               Table[ f[n], {n, 1, 16}]
%Y A020556 Cf. A020554, A014500, A020558.
%Y A020556 Sequence in context: A102923 A092586 A048363 this_sequence A007803 A034219 
               A034238
%Y A020556 Adjacent sequences: A020553 A020554 A020555 this_sequence A020557 A020558 
               A020559
%K A020556 nonn
%O A020556 0,3
%A A020556 Gilbert Labelle (gilbert(AT)lacim.uqam.ca) and Simon Plouffe (simon.plouffe(AT)gmail.com)
%E A020556 Edited by Robert G. Wilson v (rgwv(AT)rgwv.com), Apr 30 2002

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