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%I #14 Apr 28 2024 11:21:36
%S 1,0,1,0,1,1,0,3,3,5,0,13,9,20,34,0,90,46,70,170,273,0,747,312,360,
%T 680,1638,2436,0,7040,2580,2435,3570,7371,17052,23391,0,71736,24056,
%U 19800,23970,39858,85260,187128,237090,0,774738,243483,182850,193664,267813,477456,1029204,2133810,2505228
%N Triangle read by rows: T(n,k) is the number of planar tanglegrams of size n with irreducible component of size k.
%C A proper subtanglegram of a planar tanglegram is a pair of subtrees whose leaves are matched in the tanglegram, and the irreducible component of a planar tanglegram is formed by contracting each maximal proper subtanglegram into a pair of matched leaves.
%H Alexander E. Black, Kevin Liu, Alex McDonough, Garrett Nelson, Michael C. Wigal, Mei Yin, and Youngho Yoo, <a href="https://doi.org/10.1016/j.aam.2023.102550">Sampling planar tanglegrams and pairs of disjoint triangulations</a>, Advances in Applied Mathematics 149 (2023), Paper No. 102550.
%F G.f.: F(x,y) = H(F(x),y) + x*y + y^2*(F(x)^2 + F(x^2))/2 where the coefficient of x^n*y^k is the number of planar tanglegrams of size n with irreducible component of size k, F(x) is the g.f. for A349408, and H(x)/x^2 is the g.f. for A257887.
%e Triangle begins
%e 1;
%e 0, 1;
%e 0, 1, 1;
%e 0, 3, 3, 5;
%e 0, 13, 9, 20, 34;
%e 0, 90, 46, 70, 170, 273;
%e 0, 747, 312, 360, 680, 1638, 2436;
%e 0, 7040, 2580, 2435, 3570, 7371, 17052, 23391;
%e ...
%Y Cf. A349408 (diagonal), A257887 (row sums).
%K nonn,tabl
%O 1,8
%A _Kevin Liu_, Apr 01 2024
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