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A052802 E.g.f. satisfies A(x) = 1/(1 + log(1 - x*A(x))).

%I A052802 #31 Oct 31 2024 22:28:21
%S A052802 1,1,5,47,660,12414,293552,8374806,280064600,10747277832,465597887592,
%T A052802 22479948822792,1197060450322800,69699159437088960,
%U A052802 4405397142701855232,300408348609092268144,21983809533066553697280
%N A052802 E.g.f. satisfies A(x) = 1/(1 + log(1 - x*A(x))).
%C A052802 Previous name was: A simple grammar.
%H A052802 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=760">Encyclopedia of Combinatorial Structures 760</a>
%F A052802 From _Paul D. Hanna_, Aug 28 2008: (Start)
%F A052802 E.g.f. satisfies: A(x*(1 + log(1-x))) = 1/(1 + log(1-x)).
%F A052802 E.g.f. satisfies: A(x) = 1/(1 + log(1 - x*A(x))).
%F A052802 E.g.f.: A(x) = (1/x)*Series_Reversion[x + x*log(1-x)]. (End)
%F A052802 a(n)=sum(k=0..n, binomial(n+k,n)*k!*sum(j=0..k, (-1)^(n+j)/(k-j)!*sum(i=0..j, stirling1(n,j-i)/i!)))/(n+1); [_Vladimir Kruchinin_, May 09 2013]
%F A052802 a(n) ~ n^(n-1) * c^n / (sqrt(1+c) * exp(n) * (c-1)^(2*n+1)), where c = LambertW(exp(2)) = 1.5571455989976114... - _Vaclav Kotesovec_, Jan 08 2014
%F A052802 a(n) = (1/(n+1)!) * Sum_{k=0..n} (n+k)! * |Stirling1(n,k)|. - _Seiichi Manyama_, Nov 06 2023
%e A052802 E.g.f.: A(x) = 1 + x + 5*x^2/2! + 47*x^3/3! + 660*x^4/4! +... [_Paul D. Hanna_, Aug 28 2008]
%p A052802 spec := [S,{C=Cycle(B),S=Sequence(C),B=Prod(S,Z)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t A052802 CoefficientList[1/x*InverseSeries[Series[x + x*Log[1-x], {x, 0, 20}], x],x] * Range[0, 19]! (* _Vaclav Kotesovec_, Jan 08 2014 *)
%o A052802 (PARI) a(n)=n!*polcoeff((1/x)*serreverse(x+x*log(1-x +x*O(x^n))),n) \\ _Paul D. Hanna_, Aug 28 2008
%o A052802 (Maxima) a(n):=sum(binomial(n+k,n)*k!*sum((-1)^(n+j)/(k-j)!*sum(stirling1(n,j-i)/i!,i,0,j),j,0,k),k,0,n)/(n+1); /* _Vladimir Kruchinin_, May 09 2013 */
%Y A052802 Cf. A052819. [From _Paul D. Hanna_, Aug 28 2008]
%Y A052802 Cf. A052894, A198860.
%K A052802 easy,nonn
%O A052802 0,3
%A A052802 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052802 New name using e.g.f., _Vaclav Kotesovec_, Jan 08 2014