A071379 a(n) = (1/e) * Sum_{k>=0} ((k+4)!/k!)^(n-1)/k!.

1, 1, 209, 163121, 326922081, 1346634725665, 9939316337679281, 119802044788535500753, 2205421644124274191535553, 58945667435045762187763602753, 2198513228897522394476415669503377, 110833342180980170285766876408530089329

Offset: 0

Author

Karol A. Penson, May 22 2002

Keywords

• nonn

Comments

This is a Dobinski-type summation formula.

Terms quickly become gigantic: a(15) = 9142140479823239889945170786704021785456107245847570873873. a(n) appears in the process of ordering the n-th power of a product of fourth power of boson creation and fourth power of boson annihilation operators.

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_{4}(x)), where [x^n] denotes the coefficient of x^k in the Taylor series for B_{4}(x).

a(n) is row 4 of the square array representation of A090210. (End)

Links

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

Crossrefs

Cf. A000110, A020556 and A069223, when k+4 is replaced by k+1, k+2 or k+3 respectively.

Cf. A090210.

Programs

Maple

A071379 := proc(n) local r,s,i;
if n=0 then 1 else r := [seq(5,i=1..n-1)]; s := [seq(1,i=1..n-1)];
exp(-x)*24^(n-1)*hypergeom(r,s,x); round(evalf(subs(x=1,%),99)) fi end:
seq(A071379(n),n=0..10); # Peter Luschny, Mar 30 2011

Mathematica

a[n_] := Sum[FactorialPower[k, 4]^n/k!, {k, 4, Infinity}]/E; a[0] = 1; Array[a, 12, 0] (* Jean-François Alcover, Sep 01 2016 *)

PARI

default(realprecision, 500); for(n=0, 20, print1(if(n==0, 1, round(exp(-1)*sum(k=0, 500, ((k+4)!/k!)^(n-1)/k!))), ", ")) \\ G. C. Greubel, May 15 2018

Formula

a(n) = (1/e)*Sum_{k>=4} fallfac(k, 4)^n / k!, n >= 1, with fallfac(n, m) := A008279(n, m) (falling factorials). (From eq.(26) with r=4 of the Schork reference.)

E.g.f. with a(0) := 1: (1/e)*(Sum_{k>=4} e^(fallfac(k, 4)*x)/k! + 8/3). From top of p. 4656 with r=4 of the Schork reference.

Extensions

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

If it is proved that A283153 and A071379 are the same, then the entries should be merged and A283153 recycled. - N. J. A. Sloane, Mar 06 2017