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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.
5
1, 1, 1, 2, 6, 2, 5, 30, 30, 5, 14, 140, 280, 140, 14, 42, 630, 2100, 2100, 630, 42, 132, 2772, 13860, 23100, 13860, 2772, 132, 429, 12012, 84084, 210210, 210210, 84084, 12012, 429, 1430, 51480, 480480, 1681680, 2522520, 1681680, 480480, 51480, 1430
OFFSET
0,4
COMMENTS
The number of vertices is n + 1 - k.
A tree-rooted planar map is a planar map with a distinguished spanning tree.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
R. C. Mullin, On the Enumeration of Tree-Rooted Maps, Canadian Journal of Mathematics, Volume 19, 1967, pp. 174-183.
FORMULA
T(n,k) = (2*n)!/(k!*(k+1)!*(n-k)!*(n-k+1)!).
T(n,n-k) = T(n,k).
T(n, floor(n/2)) = A215288(n).
T(n,k) = A000108(n) * A001263(n+1,k+1). - Werner Schulte, Apr 04 2021
EXAMPLE
Triangle begins:
1;
1, 1;
2, 6, 2;
5, 30, 30, 5;
14, 140, 280, 140, 14;
42, 630, 2100, 2100, 630, 42;
132, 2772, 13860, 23100, 13860, 2772, 132;
429, 12012, 84084, 210210, 210210, 84084, 12012, 429;
...
MATHEMATICA
Table[(2 n)!/(k!*(k + 1)!*(n - k)!*(n - k + 1)!), {n, 0, 8}, {k, 0, n}] // Flatten (* Michael De Vlieger, Apr 06 2021 *)
PROG
(PARI)
T(n, k) = {(2*n)!/(k!*(k+1)!*(n-k)!*(n-k+1)!)}
{ for(n=0, 10, print(vector(n+1, k, T(n, k-1)))) }
CROSSREFS
Columns k=0..2 are A000108, A002457, 2*A002803.
Row sums are A005568.
Central coefficients are A342983.
Sequence in context: A268766 A214775 A196201 * A128045 A011325 A010696
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Apr 03 2021
STATUS
approved