OFFSET
0,3
COMMENTS
Previous name was: A simple grammar.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..369
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 839.
FORMULA
E.g.f.: LambertW(x/(-1+x))/x*(-1+x).
a(n) = Sum_{k=0..n} n!/k!*binomial(n-1, k-1)*(k+1)^(k-1). - Vladeta Jovovic, Sep 17 2003
a(n) ~ sqrt((exp(1)+1)*exp(1))*n^(n-1)*(1+exp(-1))^n. - Vaclav Kotesovec, Sep 29 2013
E.g.f. A(x) satisfies A(x) = exp( x*A(x)/(1-x) ) - Olivier Gérard, Dec 28 2013
E.g.f.: exp( -LambertW(-x/(1-x)) ). - Seiichi Manyama, Mar 01 2023
MAPLE
spec := [S, {C=Sequence(Z, 1 <= card), S=Set(B), B=Prod(C, S)}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);
MATHEMATICA
CoefficientList[Series[LambertW[x/(-1+x)]/x*(-1+x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 29 2013 *)
nmax = 20; A[_] = 0; Do[A[x_] = Product[Exp[x^k*A[x]], {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] * Range[0, nmax]! (* Vaclav Kotesovec, Mar 01 2024 *)
PROG
(Maxima) makelist(if n=0 then 1 else sum(n!/k!*binomial(n-1, k-1)*(k+1)^(k-1), k, 0, n), n, 0, 17); /* Bruno Berselli, May 25 2011 */
(PARI) x='x+O('x^50); Vec(serlaplace(lambertw(x/(-1+x))/x*(-1+x))) \\ G. C. Greubel, Nov 12 2017
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x/(1-x))))) \\ Seiichi Manyama, Mar 01 2023
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
EXTENSIONS
New name using e.g.f., Joerg Arndt, Sep 30 2013
STATUS
approved