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

1,8

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.

Links

Table of n, a(n) for n=1..55.

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.

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.

Example

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;
  ...

Crossrefs

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

Sequence in context: A197137 A133456 A271710 * A246005 A103786 A067462

Adjacent sequences: A371656 A371657 A371658 * A371660 A371661 A371662

Keyword

nonn,tabl

Author

Kevin Liu, Apr 01 2024

Status

approved