OFFSET
0,3
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
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) ~ 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)
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 *)
PROG
(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
(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
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
EXTENSIONS
New name, using e.g.f., by Vaclav Kotesovec, Sep 30 2013
STATUS
approved