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%I #14 Feb 08 2024 20:45:23
%S 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,
%T 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,
%U 10,38,84,190,380,340,455,42,1,1,7,10,69,92,500,506,1771,969,1428,132
%N 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.
%C The polygon prior to dissection will have n*(k-2)+2 sides.
%H Andrew Howroyd, <a href="/A370060/b370060.txt">Table of n, a(n) for n = 1..1275</a> (first 50 antidiagonals)
%H F. Harary, E. M. Palmer and R. C. Read, <a href="http://dx.doi.org/10.1016/0012-365X(75)90041-2">On the cell-growth problem for arbitrary polygons</a>, Discr. Math. 11 (1975), 371-389.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Fuss%E2%80%93Catalan_number">Fuss-Catalan number</a>
%F T(n,k) = 2*A295259(n,k) - A295222(n,k).
%F T(n,2*k+1) = A370062(n,2*k+1).
%e Array begins:
%e =============================================
%e n\k| 3 4 5 6 7 8 9 10 ...
%e ---+-----------------------------------------
%e 1 | 1 1 1 1 1 1 1 1 ...
%e 2 | 1 1 1 1 1 1 1 1 ...
%e 3 | 1 3 2 4 3 5 4 6 ...
%e 4 | 2 4 4 6 6 8 8 10 ...
%e 5 | 2 12 9 26 21 45 38 69 ...
%e 6 | 5 18 22 45 51 84 92 135 ...
%e 7 | 5 55 52 204 190 500 468 992 ...
%e 8 | 14 88 140 380 506 1008 1240 2100 ...
%e 9 | 14 273 340 1771 1950 6200 6545 15990 ...
%e ...
%o (PARI) \\ here u is Fuss-Catalan sequence with p = k-1.
%o u(n, k, r) = {r*binomial((k - 1)*n + r, n)/((k - 1)*n + r)}
%o 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)) )}
%o for(n=1, 9, for(k=3, 10, print1(T(n, k), ", ")); print);
%Y Columns k=3..6 are A208355(n-1), A124817(n-1), A369472, A370061.
%Y Cf. A070914 (rooted), A295222 (oriented), A295259 (unoriented), A369929, A370062 (achiral unrooted).
%K nonn,tabl
%O 1,9
%A _Andrew Howroyd_, Feb 08 2024