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%I #27 Jun 27 2022 08:44:32
%S 0,1,3,20,220,3424,69008,1706256,49956240,1689497376,64799254752,
%T 2778906776832,131756614920192,6843405231815424,386414606189283072,
%U 23567401521343170048,1543994621969805135360,108137637714495023354880,8062825821198926369725440
%N Expansion of e.g.f. 1/2 - (1/2)*(1+4*log(1-x))^(1/2).
%C Previous name was: A simple grammar.
%H Seiichi Manyama, <a href="/A052851/b052851.txt">Table of n, a(n) for n = 0..360</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=819">Encyclopedia of Combinatorial Structures 819</a>
%F E.g.f.: 1/2 - (1/2)*(1-4*log(-1/(-1+x)))^(1/2).
%F 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 a(n) ~ n^(n-1)/(sqrt(2)*exp(3*n/4)*(exp(1/4)-1)^(n-1/2)). - _Vaclav Kotesovec_, Sep 30 2013
%p spec := [S,{B=Cycle(Z),S=Prod(B,C),C=Sequence(S)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t 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 (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 Cf. A048287, A052886, A087138.
%K easy,nonn
%O 0,3
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E New name using e.g.f., _Vaclav Kotesovec_, Sep 30 2013
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