0,3

Generalized Bell numbers: a(n)=sum(A078739(n,k),k=2..2*n),n>=1.

G. Labelle, Counting enriched multigraphs..., Discrete Math., 217 (2000), 237-248.

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.

G. Paquin, D\'enombrement de multigraphes enrichis, M\'emoire, Math. Dept., Univ. Qu\'ebec \`a Montr\'eal, 2004.

M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

P. Blasiak, K. A. Penson and A. I. Solomon, The Boson Normal Ordering Problem and Generalized Bell Numbers

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem.

Sum( (k+2)!^n /(k+2)!*(k!^n)*exp(1)), k = 0 .. infinity, n>=1.

(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.)

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).

a(n) = Sum_{k=0..n} (-1)^k*binomial(n, k)*Bell(2*n-k). - Vladeta Jovovic (vladeta(AT)eunet.rs), May 02 2004

f[n_] := f[n] = Sum[(k + 2)!^n/((k + 2)!*(k!^n)*E), {k, 0, Infinity}]; Table[ f[n], {n, 1, 16}]

Cf. A020554, A014500, A020558.

Sequence in context: A102923 A092586 A048363 this_sequence A007803 A034219 A034238

Adjacent sequences: A020553 A020554 A020555 this_sequence A020557 A020558 A020559

nonn

Gilbert Labelle (gilbert(AT)lacim.uqam.ca) and Simon Plouffe (simon.plouffe(AT)gmail.com)

Edited by Robert G. Wilson v (rgwv(AT)rgwv.com), Apr 30 2002