A371659 Triangle read by rows: T(n,k) is the number of planar tanglegrams of size n with irreducible component of size k.

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, 680, 1638, 2436, 0, 7040, 2580, 2435, 3570, 7371, 17052, 23391, 0, 71736, 24056, 19800, 23970, 39858, 85260, 187128, 237090, 0, 774738, 243483, 182850, 193664, 267813, 477456, 1029204, 2133810, 2505228

Offset: 1

Author

Kevin Liu, Apr 01 2024

Keywords

• nonn
• tabl

Comments

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.

Examples

			Triangle begins
  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,  680, 1638,  2436;
  0, 7040, 2580, 2435, 3570, 7371, 17052, 23391;
  ...
		

Links

• Alexander E. Black, Kevin Liu, Alex McDonough, Garrett Nelson, Michael C. Wigal, Mei Yin, and Youngho Yoo, Sampling planar tanglegrams and pairs of disjoint triangulations, Advances in Applied Mathematics 149 (2023), Paper No. 102550.

Crossrefs

Cf. A349408 (diagonal), A257887 (row sums).

Formula

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.