A123510 Arises in the normal ordering of functions of a*(a+)*a, where a and a+ are the boson annihilation and creation operators, respectively.

1, 6, 42, 340, 3135, 32466, 373156, 4713192, 64877805, 966466270, 15487707246, 265617899196, 4853435351947, 94114052406570, 1930026941433480, 41728495237790416, 948549349736725401, 22613209058160908982, 564104540143144909810, 14694713818659640322340

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..440

Formula

E.g.f.: (1/(1-x)^3)*exp(x/(1-x))*LaguerreL(2,-x/(1-x)), where LaguerreL(p,y) are the Laguerre polynomials.

From Vaclav Kotesovec, Nov 13 2017: (Start)

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

a(n) ~ exp(2*sqrt(n) - n - 1/2) * n^(n + 9/4) / 2^(3/2) * (1 + 31/(48*sqrt(n))).

(End)

Mathematica

max = 16; s = (1/(1-x)^3)*Exp[x/(1-x)]*LaguerreL[2, -x/(1-x)] + O[x]^(max+1); CoefficientList[s, x]*Range[0, max]! (* Jean-François Alcover, May 23 2016 *)

Prog

(PARI) m=30; v=concat([6, 42], vector(m-2)); for(n=3, m, v[n]=2*(n+2)*v[n-1]-(n^2 - 1)*((n+2)/n)*v[n-2]); concat([1], v) \\ G. C. Greubel, May 16 2018
(Magma) I:=[6, 42]; [1] cat [n le 2 select I[n] else 2*(n+2)*Self(n-1) - (n^2 -1)*((n+2)/n)*Self(n-2): n in [1..30]]; // G. C. Greubel, May 16 2018

Crossrefs

Cf. A002720, A052852, A123511, A123512.

Sequence in context: A029588 A001725 A361547 * A265871 A132804 A232473

Adjacent sequences: A123507 A123508 A123509 * A123511 A123512 A123513

Keyword

nonn

Author

Karol A. Penson, Oct 02 2006

Extensions

a(0)=1 prepended by G. C. Greubel, Oct 31 2017

More terms from G. C. Greubel, May 16 2018

Status

approved