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A349409 Triangle read by rows: T(n,k) is the number of planar tanglegrams of size n with 0 <= k < n leaf-matched pairs. A leaf matched pair is a pair of non-leaf vertices (u,v) in the tanglegram such that the induced subtrees rooted and u and v also form a tanglegram (equivalently, the leaves in these two subtrees are matched by the matching that forms the original tanglegram).

%I A349409 #39 Jan 20 2024 16:07:05
%S A349409 1,0,1,0,1,1,0,5,4,2,0,34,28,11,3,0,273,239,102,29,6,0,2436,2283,1045,
%T A349409 325,73,11,0,23391,23475,11539,3852,968,181,23,0,237090,254309,133690,
%U A349409 47640,12923,2756,444,46,0,2505228,2864283,1605280,607743,175976,40903,7650,1085,98
%N A349409 Triangle read by rows: T(n,k) is the number of planar tanglegrams of size n with 0 <= k < n leaf-matched pairs. A leaf matched pair is a pair of non-leaf vertices (u,v) in the tanglegram such that the induced subtrees rooted and u and v also form a tanglegram (equivalently, the leaves in these two subtrees are matched by the matching that forms the original tanglegram).
%C A349409 The generating function can be proven using a generalization of the proof for A349408.
%H A349409 Andrew Howroyd, <a href="/A349409/b349409.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)
%H A349409 Kevin Liu, <a href="https://www.mat.univie.ac.at/~slc/wpapers/FPSAC2022/45.pdf">Planar Tanglegram Layouts and Single Edge Insertion</a>, Séminaire Lotharingien de Combinatoire (2022) Vol. 86, Issue B, Art. No. 45.
%F A349409 G.f.: F(x,q) = q*H(F(x,q)) + x + q*(F(x,q)^2 + F(x^2,q^2))/2 where coefficient of x^n*q^k is the number of planar tanglegrams with size n and k leaf-matched pairs, and H(x)/x^2 is the g.f. for A257887.
%e A349409 Triangle begins
%e A349409       1;
%e A349409       0,      1;
%e A349409       0,      1,     1;
%e A349409       0,      5,     4,     2;
%e A349409       0,     34,    28,    11,    3;
%e A349409       0,    273,   239,   102,   29,   6;
%e A349409       0,   2436,  2283,  1045,  325,  73,  11;
%e A349409       0,  23391, 23475, 11539, 3852, 968, 181, 23;
%e A349409       ...
%o A349409 (PARI) \\ here H(n)/x^2 is g.f. of A257887.
%o A349409 H(n)={(x - x^2 - serreverse(sum(k=0, n+1, (binomial(2*k, k)/(k+1))^2*x^(k+1)) + O(x^(n+3))))/2}
%o A349409 F(n)={my(h=H(n-2), p=O(x)); for(n=1, n, p = x + y*subst(h + O(x*x^n), x, p) + y*(p^2 + subst(subst(p,x,x^2),y,y^2))/2); p}
%o A349409 T(n)={[Vecrev(p) | p<-Vec(F(n))]}
%o A349409 {my(v=T(10)); for(n=1, #v, print(v[n]))} \\ _Andrew Howroyd_, Nov 18 2021
%Y A349409 Cf. A257887 (2nd column), A349408 (row sums), A001190 (diagonal).
%K A349409 nonn,tabl
%O A349409 1,8
%A A349409 _Kevin Liu_, Nov 16 2021