%I A069223
%S A069223 1,34,2971,513559,149670844,66653198353,42429389528215,
%T A069223 36788942253042556,41888564490333642283,60862147523250910055785,
%U A069223 110264570238241604072673394,244397290937585028603794094349,652229940568729289038518033117685,
               2067551365133160531453420400711013314,7694635622932764203876848262780670955447
%N A069223 Generalized Bell numbers.
%C A069223 a(n) occurs in the process of normal ordering of the n-th power of a 
               product of the cubes of the boson creation and boson annihilation 
               operators.
%C A069223 a(11)=110264570238241604072673394 =~ 10^26.
%D A069223 P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal 
               ordering problem, Phys. Lett. A 309 (2003) 198-205.
%D A069223 M. Schork, On the combinatorics of normal ordering bosonic operators 
               and deforming it, J. Phys. A 36 (2003) 4651-4665.
%H A069223 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 A069223 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 A069223 a(n)= sum((((k+3)!)^n)/((k+3)!*(k!)^n), k=0..infinity)/exp(1), n>=1. 
               This is a Dobinski-type summation formula.
%F A069223 a(n)= (sum(((k*(k-1)*(k-2))^n)/k!, k=3..infinity)/exp(1), n>=1. Usually 
               a(0) := 1. (From eq.(26) with r=3 of the Schork reference; rewritten 
               original eq.(25) with r=3 of the Blasiak et al. reference.)
%F A069223 E.g.f. with a(0) := 1: (sum((exp(k*(k-1)*(k-2)*x))/k!, k=3..infinity)+5/
               2)/exp(1). From top of p. 4656 with r=3 of the Schork reference.
%t A069223 f[n_] := f[n] = Sum[(k + 3)!^n/((k + 3)!*(k!^n)*E), {k, 0, Infinity}]; 
               Table[ f[n], {n, 1, 9}]
%Y A069223 Cf. A000110 and A020556, if k+3 is replaced by k+1 or k+2, respectively.
%Y A069223 Sequence in context: A056566 A160471 A138590 this_sequence A129056 A005334 
               A033511
%Y A069223 Adjacent sequences: A069220 A069221 A069222 this_sequence A069224 A069225 
               A069226
%K A069223 nonn,easy
%O A069223 1,2
%A A069223 Karol A. Penson (penson(AT)lptl.jussieu.fr), Apr 12 2002
%E A069223 Edited by Robert G. Wilson v (rgwv(AT)rgwv.com), Apr 30 2002