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A342982 Triangle read by rows: T(n,k) is the number of tree-rooted planar maps with n edges and k+1 faces, n >= 0, k = 0..n.

%I A342982 #19 Apr 30 2021 23:05:05
%S A342982 1,1,1,2,6,2,5,30,30,5,14,140,280,140,14,42,630,2100,2100,630,42,132,
%T A342982 2772,13860,23100,13860,2772,132,429,12012,84084,210210,210210,84084,
%U A342982 12012,429,1430,51480,480480,1681680,2522520,1681680,480480,51480,1430
%N A342982 Triangle read by rows: T(n,k) is the number of tree-rooted planar maps with n edges and k+1 faces, n >= 0, k = 0..n.
%C A342982 The number of vertices is n + 1 - k.
%C A342982 A tree-rooted planar map is a planar map with a distinguished spanning tree.
%H A342982 Andrew Howroyd, <a href="/A342982/b342982.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%H A342982 R. C. Mullin, <a href="https://doi.org/10.4153/CJM-1967-010-x">On the Enumeration of Tree-Rooted Maps</a>, Canadian Journal of Mathematics, Volume 19, 1967, pp. 174-183.
%F A342982 T(n,k) = (2*n)!/(k!*(k+1)!*(n-k)!*(n-k+1)!).
%F A342982 T(n,n-k) = T(n,k).
%F A342982 T(n, floor(n/2)) = A215288(n).
%F A342982 T(n,k) = A000108(n) * A001263(n+1,k+1). - _Werner Schulte_, Apr 04 2021
%e A342982 Triangle begins:
%e A342982     1;
%e A342982     1,     1;
%e A342982     2,     6,     2;
%e A342982     5,    30,    30,      5;
%e A342982    14,   140,   280,    140,     14;
%e A342982    42,   630,  2100,   2100,    630,    42;
%e A342982   132,  2772, 13860,  23100,  13860,  2772,   132;
%e A342982   429, 12012, 84084, 210210, 210210, 84084, 12012, 429;
%e A342982   ...
%t A342982 Table[(2 n)!/(k!*(k + 1)!*(n - k)!*(n - k + 1)!), {n, 0, 8}, {k, 0, n}] // Flatten (* _Michael De Vlieger_, Apr 06 2021 *)
%o A342982 (PARI)
%o A342982 T(n,k) = {(2*n)!/(k!*(k+1)!*(n-k)!*(n-k+1)!)}
%o A342982 { for(n=0, 10, print(vector(n+1, k, T(n,k-1)))) }
%Y A342982 Columns k=0..2 are A000108, A002457, 2*A002803.
%Y A342982 Row sums are A005568.
%Y A342982 Central coefficients are A342983.
%Y A342982 Cf. A001263, A215288, A269920, A342984.
%K A342982 nonn,tabl
%O A342982 0,4
%A A342982 _Andrew Howroyd_, Apr 03 2021