A093620 Values of Laguerre polynomials: a(n) = 2^n*n!*LaguerreL(n,-1/2,-2).

1, 5, 43, 499, 7193, 123109, 2430355, 54229907, 1347262321, 36833528197, 1097912385851, 35409316648435, 1227820993510153, 45528569866101989, 1797044836586213923, 75200136212985945619, 3324579846014080352225, 154797474251689486249477, 7570037033145534341015371

Offset: 0

Author

Karol A. Penson, Apr 06 2004

Keywords

• nonn

Comments

Not the same as the numerator of LaguerreL(n,-1/2,-2). - Robert G. Wilson v, Apr 08 2004

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Crossrefs

Bisection of A005425.

Programs

Magma

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(4*x/(1-2*x))/(1-2*x)^(1/2))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 11 2018

Maple

a:= proc(n) option remember; `if`(n<2, 4*n+1,
      (4*n+1)*a(n-1) -2*(n-1)*(2*n-3)*a(n-2))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Jun 22 2013

Mathematica

Table[2^n n!LaguerreL[n, -1/2, -2], {n, 0, 16}] (* Robert G. Wilson v, Apr 08 2004 *)

PARI

x='x+O('x^30); Vec(serlaplace(exp(4*x/(1-2*x))/(1-2*x)^(1/2))) \\ G. C. Greubel, May 11 2018

PARI

a(n) = 2^n*n!*pollaguerre(n, -1/2, -2); \\ Michel Marcus, Feb 05 2021

Formula

E.g.f.: exp(4*x/(1-2*x))/(1-2*x)^(1/2).

a(n) ~ n^n*2^(n-1/2)*exp(-n+2*sqrt(2*n)-1) * (1 + 5/(6*sqrt(2*n))). - Vaclav Kotesovec, Jun 22 2013

Extensions

More terms from Robert G. Wilson v, Apr 08 2004