A123512 Arises in the normal ordering of functions of a*(a+)*a, where a and a+ are the boson annihilation and creation operators, respectively.

1, 10, 105, 1190, 14630, 194796, 2798670, 43204260, 713655855, 12564061510, 234896893231, 4648313235930, 97068707038940, 2133251854548920, 49215687006553740, 1189262114277026856, 30037396074996304365

Offset: 0

Author

Karol A. Penson, Oct 02 2006

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..439

Crossrefs

Cf.: A002720, A052852, A123510, A123511.

Programs

Mathematica

CoefficientList[ Series[(1/(1 - x)^5)*Exp[x/(1 - x)]LaguerreL[4, -x/(1 - x)], {x,0,16}], x]*Range[0, 16]! (* Robert G. Wilson v, Oct 03 2006 *)

PARI

LaguerreL(n,v='x) = {
  my(x='x+O('x^(n+1)), t='t);
  subst(polcoeff(exp(-x*t/(1-x))/(1-x), n), 't, v);
};
N=17;x='x+O('x^N); Vec(serlaplace((1/(1-x)^5)*exp(x/(1-x))*LaguerreL(4,-x/(1-x)))) \\ Gheorghe Coserea, Oct 26 2017

Formula

E.g.f.: (1/(1-x)^5)*exp(x/(1-x))*LaguerreL(4,-x/(1-x)).

From Vaclav Kotesovec, Nov 13 2017: (Start)

Recurrence: n*a(n) = 2*n*(n+4)*a(n-1) - (n-1)*(n+3)*(n+4)*a(n-2).

a(n) ~ exp(2*sqrt(n)-n-1/2) * n^(n + 17/4) / (3*2^(7/2)) * (1 + 31/(48*sqrt(n))).

(End)

Extensions

a(0)=1 prepended by Gheorghe Coserea, Oct 26 2017