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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.
5
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
1,9
COMMENTS
The polygon prior to dissection will have n*(k-2)+2 sides.
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.
FORMULA
T(n,k) = 2*A295259(n,k) - A295222(n,k).
T(n,2*k+1) = A370062(n,2*k+1).
EXAMPLE
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 ...
...
PROG
(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);
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).
Sequence in context: A232096 A250030 A316456 * A375747 A344893 A338946
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Feb 08 2024
STATUS
approved