A082545 a(n) = (2*n)! * Sum_{k=0..n} binomial(n,k)/(n+k)!.

1, 3, 21, 229, 3393, 63591, 1442173, 38398641, 1174226049, 40558249963, 1561734494661, 66335687785533, 3081211226192641, 155369391396527439, 8452596370942940973, 493494408990278911561, 30777323181433121541633, 2042075395611656190239571

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..365

Formula

a(n) = n!*LaguerreL(n, n, -1).

n*a(n) + (n^3-5*n^2-n+2)*a(n-1) - 2*(n+1)*(2*n-3)*(n-1)^2*a(n-2) = 0. - Vladeta Jovovic, Jul 16 2004

E.g.f.: exp((-2*x+1-(1-4*x)^(1/2))/(2*x))/(1-4*x)^(1/2). - Mark van Hoeij, Oct 31 2011

a(n) ~ n^n*2^(2*n+1/2)/exp(n-1). - Vaclav Kotesovec, Sep 27 2012

a(n) = n!*binomial(2*n,n)*hypergeom([-n], [1+n], -1). - Peter Luschny, May 04 2017

a(n) = n! * [x^n] exp(x/(1 - x))/(1 - x)^(n+1). - Ilya Gutkovskiy, Nov 21 2017

Maple

a:= n-> simplify(n!*LaguerreL(n$2, -1)):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 27 2017

Mathematica

Table[n!*LaguerreL[n, n, -1], {n, 0, 17}] (* Jean-François Alcover, Jun 04 2019 *)

Prog

(PARI) a(n) = sum(k=0, n, k!*binomial(n, k)*binomial(2*n, k)); \\ Seiichi Manyama, May 01 2021
(PARI) a(n) = n!*pollaguerre(n, n, -1); \\ Seiichi Manyama, May 01 2021
(Magma) [Factorial(n)*Evaluate(LaguerrePolynomial(n, n), -1): n in [0..40]]; // G. C. Greubel, Aug 11 2022
(SageMath) [factorial(n)*gen_laguerre(n, n, -1) for n in (0..40)] # G. C. Greubel, Aug 11 2022

Crossrefs

Cf. A006902, A343832.

Sequence in context: A354263 A369795 A113663 * A074638 A332708 A097329

Adjacent sequences: A082542 A082543 A082544 * A082546 A082547 A082548

Keyword

nonn

Author

Vladeta Jovovic, May 11 2003

Status

approved