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%I #33 May 06 2022 20:32:34
%S 1,0,2,6,46,340,3308,36288,460752,6551424,103685232,1803956880,
%T 34247483664,704301934752,15598712592864,370149922235520,
%U 9369093828260736,251968378971718656,7174943434198029312
%N A simple grammar: sequences of pairs of cycles.
%C Stirling transform of (-1)^n*a(n)=[0,2,-6,46,-340,...] is A005359(n)=[0,2,0,24,0,...]. - _Michael Somos_, Mar 04 2004
%H Seiichi Manyama, <a href="/A052811/b052811.txt">Table of n, a(n) for n = 0..418</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=775">Encyclopedia of Combinatorial Structures 775</a>
%F a(n) = (-1)^n*Sum_{k=0..floor(n/2)} Stirling1(n, 2*k)*(2*k)!. - _Vladeta Jovovic_, Sep 22 2003
%F E.g.f.: 1/(1-log(1-x)^2).
%F a(n) = D^n(1/(1-x^2)) evaluated at x = 0, where D is the operator exp(x)*d/dx. Cf. A006252. - _Peter Bala_, Nov 25 2011
%F a(n) ~ n!/2 * exp(n)/(exp(1)-1)^(n+1). - _Vaclav Kotesovec_, Sep 30 2013
%F a(0) = 1; a(n) = 2 * Sum_{k=1..n} binomial(n,k) * |Stirling1(k,2)| * a(n-k). - _Seiichi Manyama_, May 06 2022
%p spec := [S,{B=Cycle(Z),C=Prod(B,B),S=Sequence(C)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t CoefficientList[Series[1/(1-Log[1-x]^2), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Sep 30 2013 *)
%o (PARI) a(n)=if(n<0,0,n!*polcoeff(1/(1-log(1-x)^2),n))
%o (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=2*sum(j=1, i, binomial(i, j)*abs(stirling(j, 2, 1))*v[i-j+1])); v; \\ _Seiichi Manyama_, May 06 2022
%o (PARI) a(n) = sum(k=0, n\2, (2*k)!*abs(stirling(n, 2*k, 1))); \\ _Seiichi Manyama_, May 06 2022
%o (Maxima) makelist((-1)^n*sum(stirling1(n, 2*k)*(2*k)!,k,0,floor(n/2)), n, 0, 18); \\ _Bruno Berselli_, May 25 2011
%Y Cf. A346921.
%K easy,nonn
%O 0,3
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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