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A052851 Expansion of e.g.f. 1/2 - (1/2)*(1+4*log(1-x))^(1/2).

%I A052851 #40 Sep 09 2024 09:33:21
%S A052851 0,1,3,20,220,3424,69008,1706256,49956240,1689497376,64799254752,
%T A052851 2778906776832,131756614920192,6843405231815424,386414606189283072,
%U A052851 23567401521343170048,1543994621969805135360,108137637714495023354880,8062825821198926369725440
%N A052851 Expansion of e.g.f. 1/2 - (1/2)*(1+4*log(1-x))^(1/2).
%C A052851 Previous name was: A simple grammar.
%H A052851 Seiichi Manyama, <a href="/A052851/b052851.txt">Table of n, a(n) for n = 0..360</a>
%H A052851 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=819">Encyclopedia of Combinatorial Structures 819</a>
%H A052851 <a href="/index/Res#revert">Index entries for reversions of series</a>
%F A052851 E.g.f.: 1/2 - (1/2)*(1-4*log(-1/(-1+x)))^(1/2).
%F A052851 a(n) = Sum_{k=1..n} Stirling1(n,k)*k!*C(2*k-2,k-1)/k*(-1)^(n+k). - _Vladimir Kruchinin_, May 12 2012
%F A052851 a(n) ~ n^(n-1)/(sqrt(2)*exp(3*n/4)*(exp(1/4)-1)^(n-1/2)). - _Vaclav Kotesovec_, Sep 30 2013
%F A052851 From _Seiichi Manyama_, Sep 09 2024: (Start)
%F A052851 E.g.f. satisfies A(x) = (-log(1 - x)) / (1 - A(x)).
%F A052851 E.g.f.: Series_Reversion( 1 - exp(-x * (1 - x)) ). (End)
%p A052851 spec := [S,{B=Cycle(Z),S=Prod(B,C),C=Sequence(S)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t A052851 CoefficientList[Series[1/2-1/2*(1+4*Log[1-x])^(1/2), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Sep 30 2013 *)
%o A052851 (Maxima) a(n):=sum(stirling1(n,k)*k!*binomial(2*k-2,k-1)/k*(-1)^(n+k), k,1,n); /* _Vladimir Kruchinin_, May 12 2012 */
%Y A052851 Cf. A048287, A052886, A087138.
%Y A052851 Cf. A371314, A371315.
%Y A052851 Cf. A371327, A376041, A376042.
%K A052851 easy,nonn
%O A052851 0,3
%A A052851 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052851 New name using e.g.f., _Vaclav Kotesovec_, Sep 30 2013