A052897 Expansion of e.g.f.: exp(2*x/(1-x)).

1, 2, 8, 44, 304, 2512, 24064, 261536, 3173888, 42483968, 621159424, 9841950208, 167879268352, 3065723549696, 59651093528576, 1231571119812608, 26883546193002496, 618463501807058944

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

Comments

Previous name was: A simple grammar.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 873
• N. J. A. Sloane, Transforms
• Index entries for sequences related to Laguerre polynomials

Crossrefs

Row sums of A059110.

Cf. A000262, A025168, A255806, A317364.

Programs

Magma

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(2*x/(1 - x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 15 2018

Magma

[Factorial(n)*Evaluate(LaguerrePolynomial(n, -1), -2): n in [0..25]]; // G. C. Greubel, Feb 23 2021

Maple

L := proc(n,a,x) if n=0 then 1 elif n=1 then a+1-x else (2*n+a-1-x)/n*L(n-1,a,x) - (n+a-1)/n*L(n-2,a,x) fi end: A052897 := n -> n!*L(n,-1,-2): seq(A052897(n),n=0..17); # Peter Luschny, Nov 20 2011
spec := [S,{B=Set(C),C=Sequence(Z,1 <= card),S=Prod(B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Mathematica

Range[0, 19]! CoefficientList[ Series[E^(2*x/(1 - x)), {x, 0, 19}], x] (* Zerinvary Lajos, Mar 21 2007 *)
Table[n!*LaguerreL[n, -1, -2], {n,0,30}] (* G. C. Greubel, Feb 23 2021 *)

PARI

a=Vec(exp(2*x/(1-x)));for(n=2,#a-1,a[n+1]*=n!);a \\ Charles R Greathouse IV, Nov 20 2011

Sage

[factorial(n)*gen_laguerre(n, -1, -2) for n in (0..25)] # G. C. Greubel, Feb 23 2021

Formula

E.g.f.: exp(2*x/(1-x)). - Vladeta Jovovic, Jan 04 2001

Recurrence: {a(0)=1, a(1)=2, (n^2+n)*a(n) + (-4-2*n)*a(n+1) + a(n+2)}.

LAH transform of A000079: a(n) = Sum_{k=0..n} 2^k*n!/k!*binomial(n-1, k-1). - Vladeta Jovovic, Oct 17 2003

a(n) = n!*L(n,-1,-2). - Karol A. Penson, Oct 16 2006 [Here L(n,a,x) is the n-th generalized Laguerre polynomial with parameter a, evaluated at x. L(n,a,x) is 1 if n=0, a+1-x if n=1 and otherwise (2*n+a-1-x)/n*L(n-1,a,x)-(n+a-1)/n*L(n-2,a,x). - Peter Luschny, Nov 20 2011]

a(n) ~ 2^(-1/4)*exp(2*sqrt(2*n)-n-1)*n^(n-1/4) * (1 + 7/(48*sqrt(2*n))). - Vaclav Kotesovec, Oct 09 2012, extended Dec 01 2021

E.g.f.: 1 + 2*x/((1-x)*T(0) - x), where T(k) = 4*k+1 + x^2/((4*k+3)*(1-x)^2 + x^2/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 30 2013

E.g.f.: exp(Sum_{k>=1} 2*x^k). - Vaclav Kotesovec, Mar 07 2015

a(n) = Sum_{k=0..n} binomial(n,k)*l(k)*l(n-k), where l(m) = A000262(m). - Emanuele Munarini, Aug 31 2017

Extensions

New name using e.g.f., Vaclav Kotesovec, Feb 25 2014