OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
OEIS Wiki, Exponential generating functions.
FORMULA
E.g.f.: exp(2*x/(1-x))/(1-x). - M. F. Hasler, Sep 30 2012
a(n) = n!*LaguerreL(n, -2).
a(n) = Sum_{k=0..n} 2^k*(n-k)!*binomial(n, k)^2.
E.g.f.: exp(x) * Sum_{n>=0} 2^n*x^n/n!^2 = Sum_{n>=0} a(n)*x^n/n!^2. [Paul D. Hanna, Nov 18 2011]
a(n) ~ n^(n+1/4)*exp(2*sqrt(2*n)-n-1)*2^(-3/4). - Vaclav Kotesovec, Sep 29 2012
Lim n -> infinity a(n)/(n!*BesselI(0, 2*sqrt(2*n))) = exp(-1). - Vaclav Kotesovec, Oct 12 2016
D-finite with recurrence: a(n) +(-2*n-1)*a(n-1) +(n-1)^2*a(n-2)=0. - R. J. Mathar, Feb 21 2020
MAPLE
a := proc(n) option remember: if n<1 then 1 else (2*n+1)*a(n-1) - (n-1)^2*a(n-2) fi end: 'a(n)'$n=0..17; # Zerinvary Lajos, Sep 26 2006; corrected by M. F. Hasler, Sep 30 2012
MATHEMATICA
Table[n! SeriesCoefficient[E^(2*x/(1-x))/(1-x), {x, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, May 10 2013 *)
Table[n!*LaguerreL[n, -2], {n, 0, 30}] (* G. C. Greubel, May 16 2018 *)
PROG
(PARI) A087912(n)={n!^2*polcoeff(exp(x+x*O(x^n))*sum(m=0, n, 2^m*x^m/m!^2) , n)} \\ Paul D. Hanna, Nov 18 2011
(PARI) x='x+O('x^66); Vec(serlaplace(exp(2*x/(1-x))/(1-x))) \\ Joerg Arndt, May 10 2013
(Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(2*x/(1-x))/(1-x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 16 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladeta Jovovic, Oct 18 2003
EXTENSIONS
Several minor edits by M. F. Hasler, Sep 30 2012
STATUS
approved