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A052801 A simple grammar: labeled pairs of sequences of cycles.

%I A052801 #41 Nov 19 2023 08:22:14
%S A052801 1,2,8,46,342,3108,33324,411360,5741856,89379120,1534623936,
%T A052801 28804923024,586686138384,12885385945248,303537419684064,
%U A052801 7633673997722496,204125888803996800,5782960189212871680
%N A052801 A simple grammar: labeled pairs of sequences of cycles.
%H A052801 Robert Israel, <a href="/A052801/b052801.txt">Table of n, a(n) for n = 0..417</a>
%H A052801 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=759">Encyclopedia of Combinatorial Structures 759</a>.
%F A052801 E.g.f.: 1/(-1+log(-1/(-1+x)))^2.
%F A052801 a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling1(n, k)*(k+1)!. - _Vladeta Jovovic_, Sep 21 2003
%F A052801 a(n) = D^n(1/(1-x)^2) evaluated at x = 0, where D is the operator exp(x)*d/dx. Cf. A052811. - _Peter Bala_, Nov 25 2011
%F A052801 a(n) ~ n! * n*exp(n)/(exp(1)-1)^(n+2). - _Vaclav Kotesovec_, Sep 30 2013
%F A052801 From _Anton Zakharov_, Aug 07 2016: (Start)
%F A052801 a(n) = A007840(n) + A215916(n).
%F A052801 a(n) = Sum_{k=2..n+1} k!*s(n,k) where s(n,k) is the unsigned Stirling number of the first kind, (A132393). (End)
%F A052801 a(0) = 1; a(n) = Sum_{k=1..n} (k/n + 1) * (k-1)! * binomial(n,k) * a(n-k). - _Seiichi Manyama_, Nov 19 2023
%p A052801 spec := [S,{C=Cycle(Z),B=Sequence(C),S=Prod(B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t A052801 CoefficientList[Series[1/(1+Log[1-x])^2, {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Sep 30 2013 *)
%o A052801 (Maxima) makelist(sum((-1)^(n-k)*stirling1(n, k)*(k+1)!, k, 0, n), n, 0, 17); /* _Bruno Berselli_, May 25 2011 */
%Y A052801 Cf. A215916.
%Y A052801 Cf. A007840, A354122, A354123.
%K A052801 easy,nonn
%O A052801 0,2
%A A052801 encyclopedia(AT)pommard.inria.fr, Jan 25 2000