A069223 Generalized Bell numbers: row 3 of A090210.

1, 1, 34, 2971, 513559, 149670844, 66653198353, 42429389528215, 36788942253042556, 41888564490333642283, 60862147523250910055785, 110264570238241604072673394, 244397290937585028603794094349, 652229940568729289038518033117685, 2067551365133160531453420400711013314, 7694635622932764203876848262780670955447

Offset: 0

Author

Karol A. Penson, Apr 12 2002

Keywords

• nonn
• easy

Comments

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.

a(11) = 110264570238241604072673394 =~ 10^26.

From Peter Luschny, Mar 27 2011: (Start)

Let B_{m}(x) = sum_{j>=0}(exp(j!/(j-m)!*x-1)/j!) then a(n) = n! [x^n] taylor(B_{3}(x)), where [x^n] denotes the coefficient of x^n in the Taylor series for B_{3}(x).

a(n) is row 3 of the square array representation of A090210. (End)

Links

• P. Blasiak, K. A. Penson and A. I. Solomon, The Boson Normal Ordering Problem and Generalized Bell Numbers, arXiv:quant-ph/0212072, 2002.
• P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
• P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
• P. Codara, O. M. D'Antona, and P. Hell, A simple combinatorial interpretation of certain generalized Bell and Stirling numbers, arXiv preprint arXiv:1308.1700 [cs.DM], 2013.
• P. Codara, O. M. D’Antona, and P. Hell, A simple combinatorial interpretation of certain generalized Bell and Stirling numbers, Discrete Math. 318 (2014), 53--57. MR3141626
• S.-M. Ma, T. Mansour, and M. Schork. Normal ordering problem and the extensions of the Stirling grammar, arXiv preprint arXiv:1308.0169 [math.CO], 2013.
• Toufik Mansour, Matthias Schork and Mark Shattuck, The Generalized Stirling and Bell Numbers Revisited, Journal of Integer Sequences, Vol. 15 (2012), #12.8.3.
• K. A. Penson, P. Blasiak, A. Horzela, A. I. Solomon and G. H. E. Duchamp, Laguerre-type derivatives: Dobinski relations and combinatorial identities, J. Math. Phys. 50, 083512 (2009).
• M. Riedel, Set partitions of unique elements from an n-by-m matrix where elements from the same row may not be in the same partition
• M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

Crossrefs

Cf. A000110 and A020556, if k+3 is replaced by k+1 or k+2, respectively.

Cf. A090210.

Programs

Maple

A069223 := proc(n) local r,s,i;
if n=0 then 1 else r := [seq(4,i=1..n-1)]; s := [seq(1,i=1..n-1)];
exp(-x)*6^(n-1)*hypergeom(r,s,x); round(evalf(subs(x=1,%),99)) fi end:
seq(A069223(n),n=1..15); # Peter Luschny, Mar 30 2011

Mathematica

f[n_] := f[n] = Sum[(k + 3)!^n/((k + 3)!*(k!^n)*E), {k, 0, Infinity}]; Table[ f[n], {n, 1, 9}]
a[n_] := (* row sum of A078741 *) Sum[(-1)^k*Sum[(-1)^p*((p - 2)*(p - 1)*p)^n*Binomial[k, p], {p, 3, k}]/k!, {k, 3, 3n}]; Array[a, 15] (* Jean-François Alcover, Sep 01 2015 *)

PARI

default(realprecision, 500); for(n=0, 20, print1(if(n==0, 1, round(exp(-1)*suminf(k=0, ((k+3)!)^n/( (k+3)!*(k!)^n)))), ", ")) \\ G. C. Greubel, May 15 2018

Formula

a(n) = exp(-1) * Sum_{k>=0} ((k+3)!)^n/((k+3)!*(k!)^n), n>=1. This is a Dobinski-type summation formula.

a(n) = exp(-1) * Sum_{k>=3} (k*(k-1)*(k-2))^n/k!, 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.)

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.

Extensions

Edited by Robert G. Wilson v, Apr 30 2002

a(0)=1 prepended by Alois P. Heinz, Aug 01 2016