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%I #42 Jun 02 2023 15:21:14
%S 0,1,3,23,290,5104,115374,3185972,103946688,3912527016,166884627360,
%T 7955159511672,419106982360560,24182042474691984,1516563901865906880,
%U 102717031449780063360,7472238163167018081024
%N E.g.f. A(x) = series reversion of (1-x)*(1-exp(-x)).
%C A simple grammar.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=809">Encyclopedia of Combinatorial Structures 809</a>
%H Vladimir Kruchinin, <a href="http://arxiv.org/abs/1211.3244">The method for obtaining expressions for coefficients of reverse generating functions</a>, arXiv:1211.3244 [math.CO], 2012.
%F E.g.f. satisfies: A(x) = -log(1 - x/(1-A(x))). [From Encyclopedia of Combinatorial Structures]
%F a(n) = sum(k=0..n-1, (sum(j=0..k, (sum(i=0..j, (stirling2(i+n-1,j)*C(j,j-i))/ (i+n-1)!))*(-1)^(n+j-1)/(k-j)!))*(n+k-1)!), n>0. - _Vladimir Kruchinin_, Feb 06 2012
%F a(n) ~ n^(n-1) * c^n / (sqrt(1+c) * exp(n) * (c-1)^(2*n-1)), where c = LambertW(exp(2)) = 1.5571455989976114... (see A226571). - _Vaclav Kotesovec_, Jan 08 2014
%F For n >= 1, a(n) = Sum_{k=0..n-1} Pochhammer(n, k)*|Stirling1(n, k+1)|. - _Mélika Tebni_, Jun 02 2023
%e E.g.f.: A(x) = x + 3*x^2/2! + 23*x^3/3! + 290*x^4/4! + 5104*x^5/5! +... which satisfies: A(x) = -log(1 - x/(1-A(x))).
%p spec := [S,{C=Prod(Z,B),S=Cycle(C),B=Sequence(S)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%p # second Maple program:
%p A052842 := proc (n) option remember; `if`(n = 0, 0, add(pochhammer(n, k)*abs(Stirling1(n, k+1)), k = 0..n-1)) end:
%p seq(A052842(n), n = 0..16); # _Mélika Tebni_, Jun 02 2023
%t CoefficientList[InverseSeries[Series[(-1 + E^(-x))*(x-1),{x,0,20}],x],x] * Range[0,20]! (* _Vaclav Kotesovec_, Jan 08 2014 *)
%o (PARI) {a(n)=n!*polcoeff(serreverse((1-exp(-x+O(x^(n+2))))*(1-x)),n)} /* _Paul D. Hanna_, Jun 22 2011 */
%o (Maxima) a(n):=sum((sum((sum((stirling2(i+n-1,j)*binomial(j,j-i))/(i+n-1)!,i,0,j))*(-1)^(n+j-1)/(k-j)!,j,0,k))*(n+k-1)!,k,0,n-1); /* _Vladimir Kruchinin_, Feb 06 2012 */
%Y Cf. A226571.
%K easy,nonn
%O 0,3
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E Name from a comment by _Paul D. Hanna_, Jun 22 2011
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