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%I #32 Dec 29 2020 20:41:36
%S 1,1,1,2,4,31,243,3532,62810,1390718,36080361,1076477512,36281518847,
%T 1363869480379,56587508558171,2569141702825037,126714642738385906,
%U 6747643861563535720,385875940575529343271,23588199955061659841248,1535037278334227258123709,105961521687913311720698169
%N Number of unrooted binary ordered tanglegrams of size n.
%C Binary tanglegrams are pairs of bifurcating (degree 3 internal node) trees with a bijection between the leaves of the trees. Two tanglegrams are isomorphic if there is an isomorphism between the trees that preserves the bijection. Unrooted means that the tanglegram is composed of unrooted trees, and ordered means that the trees are considered as an ordered pair.
%H Andrew Howroyd, <a href="/A259115/b259115.txt">Table of n, a(n) for n = 1..50</a>
%H S. C. Billey, M. Konvalinka, and F. A. Matsen IV, <a href="http://arxiv.org/abs/1507.04976">On the enumeration of tanglegrams and tangled chains</a>, arXiv:1507.04976 [math.CO], 2015.
%H Ira M. Gessel, <a href="http://arxiv.org/abs/1509.03867">Counting tanglegrams with species</a>, arXiv:1509.03867 [math.CO], (13-September-2015)
%H F. A. Matsen IV, S. C. Billey, D. A. Kas, and M. Konvalinka, <a href="http://arxiv.org/abs/1507.04784">Tanglegrams: a reduction tool for mathematical phylogenetics</a>, arXiv:1507.04784 [q-bio.PE], 2015.
%H Frederick A. Matsen, <a href="https://github.com/matsengrp/tangle">Sage/GAP4 Code for generating tanglegrams</a>
%o (PARI) \\ See links in A339645 for combinatorial species functions.
%o rootedBinTrees(n)={my(v=vector(n)); v[1]=sv(1); for(n=2, n, v[n]=(sum(j=1, n-1, v[j]*v[n-j]) + if(n%2, 0, sRaiseCI(v[n/2], n/2, 2)))/2); x*Ser(v)}
%o cycleIndexSeries(n)={my(g=rootedBinTrees(n), u = g + (sRaise(g,3) - g^3)/3); sCartProd(u,u)}
%o NumUnlabeledObjsSeq(cycleIndexSeries(12)) \\ _Andrew Howroyd_, Dec 24 2020
%Y Cf. A258620 (tanglegrams), A259114, A259116, A258486 (tangled chains), A258487, A258488, A258489.
%K nonn
%O 1,4
%A _Frederick A. Matsen IV_, Jun 18 2015
%E More terms from _Ira M. Gessel_, Jul 19 2015
%E Terms a(15) and beyond from _Andrew Howroyd_, Dec 24 2020
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