A370060 Array read by antidiagonals: T(n,k) is the number of achiral dissections of a polygon into n k-gons by nonintersecting diagonals rooted at a cell, n >= 1, k >= 3.

1, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 2, 4, 2, 1, 1, 4, 4, 12, 5, 1, 1, 3, 6, 9, 18, 5, 1, 1, 5, 6, 26, 22, 55, 14, 1, 1, 4, 8, 21, 45, 52, 88, 14, 1, 1, 6, 8, 45, 51, 204, 140, 273, 42, 1, 1, 5, 10, 38, 84, 190, 380, 340, 455, 42, 1, 1, 7, 10, 69, 92, 500, 506, 1771, 969, 1428, 132

Offset: 1

Author

Andrew Howroyd, Feb 08 2024

Keywords

• nonn
• tabl

Comments

The polygon prior to dissection will have n*(k-2)+2 sides.

Examples

			Array begins:
=============================================
n\k|  3   4   5    6    7    8    9    10 ...
---+-----------------------------------------
1  |  1   1   1    1    1    1    1     1 ...
2  |  1   1   1    1    1    1    1     1 ...
3  |  1   3   2    4    3    5    4     6 ...
4  |  2   4   4    6    6    8    8    10 ...
5  |  2  12   9   26   21   45   38    69 ...
6  |  5  18  22   45   51   84   92   135 ...
7  |  5  55  52  204  190  500  468   992 ...
8  | 14  88 140  380  506 1008 1240  2100 ...
9  | 14 273 340 1771 1950 6200 6545 15990 ...
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals)
• F. Harary, E. M. Palmer and R. C. Read, On the cell-growth problem for arbitrary polygons, Discr. Math. 11 (1975), 371-389.
• Wikipedia, Fuss-Catalan number

Crossrefs

Columns k=3..6 are A208355(n-1), A124817(n-1), A369472, A370061.

Cf. A070914 (rooted), A295222 (oriented), A295259 (unoriented), A369929, A370062 (achiral unrooted).

Programs

PARI

\\ here u is Fuss-Catalan sequence with p = k-1.
u(n, k, r) = {r*binomial((k - 1)*n + r, n)/((k - 1)*n + r)}
T(n, k) = {if(k%2, if(n%2, u((n-1)/2, k, (k-1)/2), u(n/2-1, k, (k-1))), if(n%2, u((n-1)/2, k, k/2+1), u(n/2-1, k, k)) )}
for(n=1, 9, for(k=3, 10, print1(T(n, k), ", ")); print);

Formula

T(n,k) = 2*A295259(n,k) - A295222(n,k).

T(n,2*k+1) = A370062(n,2*k+1).