A087860 Expansion of e.g.f.: (1-exp(x/(x-1)))/(1-x).

0, 1, 3, 10, 39, 176, 905, 5244, 34111, 250480, 2108529, 20751380, 241315151, 3282366504, 50786289385, 865850559196, 15856276032255, 306665879765984, 6199863566817761, 130237717066988580, 2832527601333186319

Offset: 0

Author

Vladeta Jovovic, Oct 25 2003

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..448

Crossrefs

Cf. A009940, A070779.

Programs

Magma

I:=[1,3,10]; [0] cat [n le 3 select I[n] else 3*(n-1)*Self(n-1) - (n-1)*(3*n-5)*Self(n-2) +(n-1)*(n-2)^2*Self(n-3): n in [1..30]];

Mathematica

With[{nn=20},CoefficientList[Series[(1-Exp[x/(x-1)])/(1-x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Nov 27 2015 *)

PARI

x='x+O('x^30); concat([0], Vec(serlaplace((1-exp(x/(x-1)))/(1-x)))) \\ G. C. Greubel, Feb 06 2018

Formula

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

a(n) = 3*(n-1)*a(n-1) - (n-1)*(3*n - 5)*a(n-2) + (n-2)^2*(n-1)*a(n-3). - Vaclav Kotesovec, Nov 13 2017

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) * (1 - BesselJ(0,2*sqrt(x))). - Ilya Gutkovskiy, Jul 17 2020

a(n) = n*n!*hypergeom([1 - n, 1], [2, 2], 1). - Peter Luschny, May 10 2021

a(n) ~ n! * (1 - exp(1/2)*cos(2*sqrt(n) - Pi/4) / (sqrt(Pi) * n^(1/4))). - Vaclav Kotesovec, May 10 2021

Extensions

Definition clarified by Harvey P. Dale, Nov 27 2015