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%I #10 Feb 18 2018 20:16:40
%S 0,1,1,7,68,941,16657,360151,9197036,270900242,9041240104,
%T 337195959574,13898017639838,627328651766168,30776662410513268,
%U 1630608894822320320,92788669297928611880,5644035534941116506704
%N Number of labeled PQ-trees with n leaves.
%C A PQ-tree is a rooted tree with P-type internal nodes that have at least 3 children that are reversibly ordered (the reverse of the order is equivalent to the order) and Q-type internal nodes that have at least 2 unordered children.
%D F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 242 (3.3.91).
%H Christian G. Bower, <a href="/A136629/b136629.txt">Table of n, a(n) for n = 0..127</a>
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F E.g.f. satisfies A(x) = x + 1/(2-2A(x)) + exp(A(x)) - A(x)^2/2 - 3/2*(A(x)+1).
%F a(n) ~ 2^n * n^(n-1) * sqrt((2 - 10*s + 15*s^2 - 8*s^3 + s^5)/(-4 + 6*s - 3*s^2 - 3*s^3 + 2*s^4)) * ((s-1)^2/(-2 + 8*s - 7*s^2 + s^3 + s^4))^n / exp(n), where s = 0.4037320373976420090487567872... is the root of the equation exp(s) + 1/(2*(s-1)^2) = 5/2 + s. - _Vaclav Kotesovec_, Jan 08 2014
%t CoefficientList[InverseSeries[Series[-E^x + (-2+x*(-2+x*(4+x)))/(2*(-1+x)),{x,0,20}],x],x] * Range[0,20]! (* _Vaclav Kotesovec_, Jan 08 2014 *)
%o (PARI) read("transforms_pari.txt"); {pql(A) = A = trv_chain_l(A)+trv_exp(A)-opv_mul_egf(A,A)/2-2*A; A[1]=0; A} {apql(n) = local(SX,SY); SY = SX = [0,1]; for(i=1,n,SY=concat(SY,0);SX=concat(SX,0);SY=SX+pql(SY)); SY} A136629(n) = apql(min(1,n-1))[n+1]
%K nonn
%O 0,4
%A _Christian G. Bower_, Jan 14 2008
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