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A052871 Expansion of e.g.f. -LambertW(x/(-1+x)).

%I A052871 #29 Sep 10 2024 08:03:52
%S A052871 0,1,4,27,268,3585,60846,1255471,30535912,855688833,27148954330,
%T A052871 962037575631,37659124454700,1613921425656865,75156944627712598,
%U A052871 3778932799275876495,204039148080188427856,11774630933193827543553
%N A052871 Expansion of e.g.f. -LambertW(x/(-1+x)).
%C A052871 Previous name was: A simple grammar.
%H A052871 G. C. Greubel, <a href="/A052871/b052871.txt">Table of n, a(n) for n = 0..369</a>
%H A052871 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=842">Encyclopedia of Combinatorial Structures 842</a>
%H A052871 <a href="/index/Res#revert">Index entries for reversions of series</a>
%F A052871 E.g.f.: -LambertW(x/(-1+x))
%F A052871 a(n) = Sum_{k=1..n} (n!/k!)*binomial(n-1, k-1)*k^(k-1). - _Vladeta Jovovic_, Sep 17 2003
%F A052871 a(n) ~ (1+exp(-1))^(n+1/2)*n^(n-1). - _Vaclav Kotesovec_, Sep 30 2013
%F A052871 From _Seiichi Manyama_, Sep 10 2024: (Start)
%F A052871 E.g.f. A(x) satisfies A(x) = x * (A(x) + exp(A(x))).
%F A052871 E.g.f.: Series_Reversion( x / (x + exp(x)) ). (End)
%p A052871 spec := [S,{C=Sequence(Z,1 <= card),B=Set(S),S=Prod(B,C)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t A052871 CoefficientList[Series[-LambertW[x/(-1+x)], {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Sep 30 2013 *)
%o A052871 (Maxima) makelist(sum((n!/k!)*binomial(n-1, k-1)*k^(k-1), k, 1, n), n, 0, 17);  /* _Bruno Berselli_, May 25 2011 */
%o A052871 (PARI) x='x+O('x^50); concat([0], Vec(serlaplace(- lambertw(x/(-1+x))) )) \\ _G. C. Greubel_, Nov 08 2017
%Y A052871 Cf. A008297, A060356.
%K A052871 easy,nonn
%O A052871 0,3
%A A052871 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052871 New name using e.g.f., _Vaclav Kotesovec_, Sep 30 2013