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A370062
Array read by antidiagonals: T(n,k) is the number of achiral dissections of a polygon into n k-gons by nonintersecting diagonals, n >= 1, k >= 3.
8
1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 3, 2, 1, 1, 3, 4, 7, 5, 1, 1, 3, 5, 9, 12, 5, 1, 1, 4, 6, 18, 22, 30, 14, 1, 1, 4, 7, 21, 35, 52, 55, 14, 1, 1, 5, 8, 34, 51, 136, 140, 143, 42, 1, 1, 5, 9, 38, 70, 190, 285, 340, 273, 42, 1, 1, 6, 10, 55, 92, 368, 506, 1155, 969, 728, 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*A295260(n,k) - A295224(n,k).
T(n,2*k+1) = A370060(n,2*k+1).
T(n,2*k) = A369929(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 2 2 3 3 4 4 5 ...
4 | 2 3 4 5 6 7 8 9 ...
5 | 2 7 9 18 21 34 38 55 ...
6 | 5 12 22 35 51 70 92 117 ...
7 | 5 30 52 136 190 368 468 775 ...
8 | 14 55 140 285 506 819 1240 1785 ...
9 | 14 143 340 1155 1950 4495 6545 12350 ...
...
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(n%2, u((n-1)/2, k, k\2), if(k%2, u(n/2-1, k, k-1), u(n/2, k, 1))))}
for(n=1, 9, for(k=3, 10, print1(T(n, k), ", ")); print);
CROSSREFS
Columns are A208355(n-1), A047749 (k=4), A369472 (k=5), A143546 (k=6), A143547 (k=8), A143554 (k=10), A192893 (k=12).
Cf. A070914 (rooted), A295224 (oriented), A295260 (unoriented), A369929, A370060 (achiral rooted at cell).
Sequence in context: A304945 A176298 A259575 * A169623 A245558 A011847
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Feb 08 2024
STATUS
approved