A052838 Expansion of e.g.f.: (exp(x/(1-x)) - 1)^2.

0, 0, 2, 18, 158, 1510, 15962, 186270, 2385182, 33290862, 503277242, 8193803926, 142938943886, 2659770747270, 52581058479770, 1100423513438766, 24302677755662654, 564770268904566238

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

Previous name was: A simple grammar.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..250
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 805
• Index entries for sequences related to Laguerre polynomials

Crossrefs

Cf. A000262, A052897.

Programs

Magma

[0] cat [2*Factorial(n)*(&+[Binomial(n-1,j)*(2^j-1)/Factorial(j+1): j in [0..n-1]]) : n in [1..25]]; // G. C. Greubel, Feb 23 2021

Maple

spec := [S,{C=Sequence(Z,1 <= card),B=Set(C,1 <= card),S=Prod(B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Mathematica

CoefficientList[Series[(E^(x/(1-x))-1)^2, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2013 *)
Table[n!*(LaguerreL[n, -1, -2] - 2* LaguerreL[n, -1, -1]) + Boole[n==0], {n,0,20}] (* G. C. Greubel, Feb 23 2021 *)

Sage

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

Formula

E.g.f.: (exp(-x/(-1+x)) - 1)^2.

Recurrence: n*(2 +5*n +4*n^2 +n^3)*a(n) - (18 +35*n +21*n^2 +4*n^3)*a(n+1) +2*(19 +15*n +3*n^2)*a(n+2) - (13 +4*n)*a(n+3) + a(n+4) = 0, with a(1)=0, a(0)=0, a(2)=2, a(3)=18.

From Vaclav Kotesovec, Sep 30 2013: (Start)

a(n) = A052897(n) - 2*A000262(n) for n > 0.

a(n) ~ 2^(-1/4)*exp(2*sqrt(2*n)-n-1)*n^(n-1/4). (End)

From G. C. Greubel, Feb 23 2021: (Start)

a(n) = 2 * n! * Sum_{j=0..n-1} binomial(n-1, j)*(2^j -1)/(j+1)!.

a(n) = n! * (LaguerreL(n, -1, -2) - 2*LaguerreL(n, -1, -1)) + [n=0]. (End)

Extensions

New name, using e.g.f., by Vaclav Kotesovec, Sep 30 2013