A342983 Number of tree-rooted planar maps with n+1 vertices and n+1 faces.

1, 6, 280, 23100, 2522520, 325909584, 47117214144, 7383099180600, 1229149289511000, 214527522662653200, 38887279926227853120, 7271332144993605081120, 1395321310426879365566400, 273697641660657106322640000, 54708248601655917595233984000

Offset: 0

Author

Andrew Howroyd, Apr 03 2021

Keywords

• nonn

Comments

The number of edges is 2*n.

Also, a(n) is the number of discrete walks that start and stop at the origin, never pass below the x-axis nor to the left of the y-axis, and, in any order, have n steps that increment x, n steps that decrement x, n steps that increment y, and n steps that decrement y. It is in this sense a way to generalize the 2n-step one-dimensional walks counted by A000108 to a count in two dimensions. Proof: There are A001448(n) ways to interleave two length-2n Dyck words (A000108(n)^2) - Lee A. Newberg, Nov 17 2023

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200

Crossrefs

Central coefficients of A342982.

Even bisection of A215288.

Cf. A000108, A001246, A001448.

Programs

PARI

a(n) = {(4*n)!/(n!*(n+1)!)^2}

Formula

a(n) = (4*n)!/(n!*(n+1)!)^2.

a(n) = A000108(n)^2 * A001448(n) = A001246(n) * A001448(n). - Alois P. Heinz, Aug 02 2023