A136629 Number of labeled PQ-trees with n leaves.

0, 1, 1, 7, 68, 941, 16657, 360151, 9197036, 270900242, 9041240104, 337195959574, 13898017639838, 627328651766168, 30776662410513268, 1630608894822320320, 92788669297928611880, 5644035534941116506704

Offset

0,4

Comments

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.

References

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 242 (3.3.91).

Links

Christian G. Bower, Table of n, a(n) for n = 0..127

Index entries for sequences related to rooted trees

Formula

E.g.f. satisfies A(x) = x + 1/(2-2A(x)) + exp(A(x)) - A(x)^2/2 - 3/2*(A(x)+1).

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

Mathematica

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 *)

Prog

(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]

Crossrefs

Sequence in context: A328046 A371392 A306386 * A197525 A133697 A224758

Adjacent sequences: A136626 A136627 A136628 * A136630 A136631 A136632

Keyword

nonn

Author

Christian G. Bower, Jan 14 2008

Status

approved