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

%I A342983 #17 Nov 18 2023 08:36:49
%S A342983 1,6,280,23100,2522520,325909584,47117214144,7383099180600,
%T A342983 1229149289511000,214527522662653200,38887279926227853120,
%U A342983 7271332144993605081120,1395321310426879365566400,273697641660657106322640000,54708248601655917595233984000
%N A342983 Number of tree-rooted planar maps with n+1 vertices and n+1 faces.
%C A342983 The number of edges is 2*n.
%C A342983 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
%H A342983 Andrew Howroyd, <a href="/A342983/b342983.txt">Table of n, a(n) for n = 0..200</a>
%F A342983 a(n) = (4*n)!/(n!*(n+1)!)^2.
%F A342983 a(n) = A000108(n)^2 * A001448(n) = A001246(n) * A001448(n). - _Alois P. Heinz_, Aug 02 2023
%o A342983 (PARI) a(n) = {(4*n)!/(n!*(n+1)!)^2}
%Y A342983 Central coefficients of A342982.
%Y A342983 Even bisection of A215288.
%Y A342983 Cf. A000108, A001246, A001448.
%K A342983 nonn
%O A342983 0,2
%A A342983 _Andrew Howroyd_, Apr 03 2021