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A052813 Expansion of e.g.f.: LambertW(log(1-x))/log(1-x).

%I A052813 #44 Aug 13 2024 01:52:38
%S A052813 1,1,4,27,260,3280,51414,965762,21175496,531317520,15021531840,
%T A052813 472654558992,16385500397496,620612495460048,25500923655523848,
%U A052813 1129909190812470840,53705490284841870144,2725878142900911376896
%N A052813 Expansion of e.g.f.: LambertW(log(1-x))/log(1-x).
%C A052813 Previous name was: A simple grammar.
%C A052813 Given e.g.f. A(x), log(A(x)) = -log(1-x)*A(x) equals e.g.f. of A052807. - _Paul D. Hanna_, Jul 19 2006
%H A052813 G. C. Greubel, <a href="/A052813/b052813.txt">Table of n, a(n) for n = 0..375</a>
%H A052813 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=778">Encyclopedia of Combinatorial Structures 778</a>
%F A052813 a(n) = Sum_{k=0..n} |Stirling1(n, k)|*(k+1)^(k-1). - _Vladeta Jovovic_, Nov 12 2003
%F A052813 E.g.f. A(x) satisfies: A(x) = 1/(1-x)^A(x). - _Paul D. Hanna_, Jul 19 2006
%F A052813 E.g.f.: Sum_{n>=0} (n+1)^(n-1)*(-log(1-x))^n/n!. - _Paul D. Hanna_, Jun 22 2009
%F A052813 E.g.f. satisfies: A(x) = Sum_{n>=0} x^n/n! * Sum_{k=0..n} |Stirling1(n, k)|*A(x)^k. - _Paul D. Hanna_, Jan 16 2013
%F A052813 a(n) ~ n^(n-1) * exp(3/2+n*exp(-1)-n) / (exp(exp(-1))-1)^(n-1/2). - _Vaclav Kotesovec_, Sep 30 2013
%F A052813 E.g.f. satisfies: A(x) = Sum_{n>=0} x^n/n! * Product_{k=0..n-1} (A(x) + k). - _Paul D. Hanna_, Oct 26 2015
%e A052813 E.g.f.: A(x) = 1 + x + 4*x^2/2! + 27*x^3/3! + 260*x^4/4! +...
%e A052813 Log(A(x))/A(x) = -log(1-x) = x + 1/2*x^2 + 1/3*x^3 + 1/4*x^4 +...
%p A052813 spec := [S,{C=Cycle(Z),S=Set(B),B=Prod(C,S)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t A052813 CoefficientList[Series[1/Log[1-x]*LambertW[Log[1-x]], {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Sep 30 2013 *)
%o A052813 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1/(1-x+x*O(x^n))^A);n!*polcoeff(A,n)} \\ _Paul D. Hanna_, Jul 19 2006
%o A052813 (PARI) {a(n)=n!*polcoeff(sum(m=0,n,(m+1)^(m-1)/m!*(-log(1-x+x*O(x^n)))^m),n)} \\ _Paul D. Hanna_, Jun 22 2009
%o A052813 (PARI) {a(n) = my(A=1+x); for(i=1,n, A = sum(m=0,n, x^m/m! * prod(k=0,m-1,A + k) +x*O(x^n)) ); n!*polcoeff(A,n)}
%o A052813 for(n=0,20, print1(a(n),", ")) \\ _Paul D. Hanna_, Oct 26 2015
%o A052813 (PARI) x='x+O('x^30); Vec(serlaplace((1/log(1-x))*lambertw(log(1-x)))) \\ _G. C. Greubel_, Feb 19 2018
%Y A052813 Cf. A052807 (log(A(x))).
%K A052813 easy,nonn
%O A052813 0,3
%A A052813 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052813 New name using e.g.f. from _Vaclav Kotesovec_, Sep 30 2013