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).

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, 325, 73, 11, 0, 23391, 23475, 11539, 3852, 968, 181, 23, 0, 237090, 254309, 133690, 47640, 12923, 2756, 444, 46, 0, 2505228, 2864283, 1605280, 607743, 175976, 40903, 7650, 1085, 98

Offset: 1

Author

Kevin Liu, Nov 16 2021

Keywords

• nonn
• tabl

Comments

The generating function can be proven using a generalization of the proof for A349408.

Examples

			Triangle begins
      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,  325,  73,  11;
      0,  23391, 23475, 11539, 3852, 968, 181, 23;
      ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)
• Kevin Liu, Planar Tanglegram Layouts and Single Edge Insertion, Séminaire Lotharingien de Combinatoire (2022) Vol. 86, Issue B, Art. No. 45.

Crossrefs

Cf. A257887 (2nd column), A349408 (row sums), A001190 (diagonal).

Programs

PARI

\\ here H(n)/x^2 is g.f. of A257887.
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}
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}
T(n)={[Vecrev(p) | p<-Vec(F(n))]}
{my(v=T(10)); for(n=1, #v, print(v[n]))} \\ Andrew Howroyd, Nov 18 2021

Formula

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.