A090209 Generalized Bell numbers (from (5,5)-Stirling2 array A090216).

1, 1, 1546, 12962661, 363303011071, 25571928251231076, 3789505947767235111051, 1049433111253356296672432821, 498382374325731085522315594481036, 380385281554629647028734545622539438171, 443499171330317702437047276255605780991365151

Offset: 0

Author

Wolfdieter Lang, Dec 01 2003

Keywords

• nonn
• easy

Comments

Contribution 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_{5}(x)), where [x^n] denotes the coefficient of x^k in the Taylor series for B_{5}(x).

a(n) is row 5 of the square array representation of A090210. (End)

References

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..115
• P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem., arXiv:quant-ph/0402027, Phys. Lett. A 309 (3-4) (2003) 198-205
• 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).

Crossrefs

Cf. (Generalized) Bell numbers from (m,m)-Stirling2 array: A000110 (m=1), A020556 (m=2), A069223 (m=3), A071379 (m=4). Triangle A090210.

Programs

Maple

A071379 := proc(n) local r,s,i;
if n=0 then 1 else r := [seq(6,i=1..n-1)]; s := [seq(1,i=1..n-1)];
exp(-x)*5!^(n-1)*hypergeom(r,s,x); round(evalf(subs(x=1,%),99)) fi end:
seq(A071379(n),n=0..8); # Peter Luschny, Mar 30 2011

Mathematica

fallfac[n_, k_] := Pochhammer[n-k+1, k]; a[n_, k_] := (((-1)^k)/k!)*Sum[((-1)^p)*Binomial[k, p]*fallfac[p, 5]^n, {p, 5, k}]; a[0] = 1; a[n_] := Sum[a[n, k], {k, 5, 5*n}];  Table[a[n], {n, 0, 8}] (* Jean-François Alcover, Mar 05 2014 *)

Formula

a(n) = Sum_{k=5..5*n} A090216(n, k), n>=1. a(0) := 1.

a(n) = Sum_{k >=5} (fallfac(k, 5)^n)/k!/exp(1), n>=1, a(0) := 1. From eq.(26) with r=5 of the Schork reference.

E.g.f. with a(0) := 1: (sum((exp(fallfac(k, 5)*x))/k!, k=5..infinity)+ A000522(4)/4!)/exp(1). From the top of p. 4656 with r=5 of the Schork reference.

Extensions

If it is proved that A283154 and A090209 are the same, then the entries should be merged and A283154 recycled. - N. J. A. Sloane, Mar 06 2017