A052868 - OEIS

%I #38 Mar 01 2024 07:53:57

%S 1,1,5,40,449,6556,118507,2561518,64540625,1859206600,60309007091,

%T 2176222795594,86488677518905,3754431762036892,176771908657345835,

%U 8973513955735900246,488586200931213192353,28404347922603101834512

%N Expansion of e.g.f. LambertW(x/(-1+x))/x*(-1+x).

%C Previous name was: A simple grammar.

%H G. C. Greubel, <a href="/A052868/b052868.txt">Table of n, a(n) for n = 0..369</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=839">Encyclopedia of Combinatorial Structures 839</a>.

%F E.g.f.: LambertW(x/(-1+x))/x*(-1+x).

%F a(n) = Sum_{k=0..n} n!/k!*binomial(n-1, k-1)*(k+1)^(k-1). - _Vladeta Jovovic_, Sep 17 2003

%F a(n) ~ sqrt((exp(1)+1)*exp(1))*n^(n-1)*(1+exp(-1))^n. - _Vaclav Kotesovec_, Sep 29 2013

%F E.g.f. A(x) satisfies  A(x) = exp( x*A(x)/(1-x) ) - _Olivier Gérard_, Dec 28 2013

%F E.g.f.: exp( -LambertW(-x/(1-x)) ). - _Seiichi Manyama_, Mar 01 2023

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

%t CoefficientList[Series[LambertW[x/(-1+x)]/x*(-1+x), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Sep 29 2013 *)

%t 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 *)

%o (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 */

%o (PARI) x='x+O('x^50); Vec(serlaplace(lambertw(x/(-1+x))/x*(-1+x))) \\ _G. C. Greubel_, Nov 12 2017

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x/(1-x))))) \\ _Seiichi Manyama_, Mar 01 2023

%Y Cf. A352410, A370436.

%K easy,nonn

%O 0,3

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

%E New name using e.g.f., _Joerg Arndt_, Sep 30 2013