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

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

Offset

0,3

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).

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.

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 *)

Crossrefs

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

Sequence in context: A249008 A249473 A246899 * A283154 A157347 A255356

Adjacent sequences: A090206 A090207 A090208 * A090210 A090211 A090212

Keyword

nonn,easy

Author

Wolfdieter Lang, Dec 01 2003

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

Status

approved