A129683 Expansion of (1/(1-2x))*exp(2x/(1-2x)).

1, 4, 28, 272, 3344, 49472, 852928, 16758016, 369082624, 8996922368, 240294124544, 6974172532736, 218457925292032, 7342688736329728, 263513011737051136, 10054112734940561408, 406301504472849907712, 17333090863574658842624, 778298003006760943353856

Offset: 0

Author

Paul Barry, Apr 28 2007

Keywords

• easy
• nonn

Comments

a(n) = 2^n*A002720(n).

Programs

Maple

a := n -> 2^n*n!*LaguerreL(n, -1):
seq(simplify(a(n)), n=0..18); # Peter Luschny, Jan 19 2020

Mathematica

Table[2^n*n!*Sum[Binomial[n,k]/k!,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Sep 26 2013 *)
CoefficientList[Series[(1/(1-2*x))*E^(2*x/(1-2*x)), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 26 2013 *)

Formula

E.g.f.: (1/(1-2x))*exp(2x/(1-2x)). - corrected by Vaclav Kotesovec, Sep 26 2013

a(n) = 2^n*n!*sum{k=0..n, C(n,k)/k!}.

Conjecture: a(n) -4*n*a(n-1) +4*(n-1)^2*a(n-2)=0. - R. J. Mathar, Nov 14 2011

a(n) ~ 2^(n-1/2)*n^(n+1/4)*exp(2*sqrt(n)-n-1/2) * (1 + 31/(48*sqrt(n))). - Vaclav Kotesovec, Sep 26 2013

a(n) = 2^n*n!*LaguerreL(n, -1). - Peter Luschny, Jan 19 2020

Extensions

Error in definition corrected by Vaclav Kotesovec, Sep 26 2013