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%I #14 Feb 24 2024 00:48:47
%S 1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,2,3,1,1,1,1,3,3,6,1,1,1,1,3,5,7,
%T 10,1,1,1,1,4,5,16,12,20,1,1,1,1,4,7,18,31,30,35,1,1,1,1,5,7,31,35,
%U 102,55,70,1,1,1,1,5,9,34,64,136,213,143,126,1
%N Array read by antidiagonals: T(n,k) is the number of achiral noncrossing partitions composed of n blocks of size k.
%C T(n,2*k-1) is the number of achiral noncrossing k-gonal cacti with n polygons.
%H Andrew Howroyd, <a href="/A369929/b369929.txt">Table of n, a(n) for n = 0..1325</a> (first 51 antidiagonals)
%H Michel Bousquet and Cédric Lamathe, <a href="https://doi.org/10.46298/dmtcs.420">On symmetric structures of order two</a>, Discrete Math. Theor. Comput. Sci. 10 (2008), 153-176. See Table 1.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Fuss%E2%80%93Catalan_number">Fuss-Catalan number</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Noncrossing_partition">Noncrossing partition</a>.
%F T(n,k) = 2*A303929(n,k) - A303694(n,k).
%F T(n,2*k-1) = 2*A361239(n,k) - A361236(n,k).
%e Array begins:
%e ===============================================
%e n\k| 1 2 3 4 5 6 7 8 9 ...
%e ---+-------------------------------------------
%e 0 | 1 1 1 1 1 1 1 1 1 ...
%e 1 | 1 1 1 1 1 1 1 1 1 ...
%e 2 | 1 1 1 1 1 1 1 1 1 ...
%e 3 | 1 2 2 3 3 4 4 5 5 ...
%e 4 | 1 3 3 5 5 7 7 9 9 ...
%e 5 | 1 6 7 16 18 31 34 51 55 ...
%e 6 | 1 10 12 31 35 64 70 109 117 ...
%e 7 | 1 20 30 102 136 296 368 651 775 ...
%e 8 | 1 35 55 213 285 663 819 1513 1785 ...
%e 9 | 1 70 143 712 1155 3142 4495 9304 12350 ...
%e ...
%o (PARI) \\ u(n,k,r) are Fuss-Catalan numbers.
%o u(n,k,r) = {r*binomial(k*n + r, n)/(k*n + r)}
%o e(n,k) = {sum(j=0, n\2, u(j, k, 1+(n-2*j)*k/2))}
%o T(n, k)={if(n==0, 1, if(k%2, if(n%2, 2*u(n\2, k, (k+1)/2), u(n/2, k, 1) + u(n/2-1, k, k)), e(n, k) + if(n%2, u(n\2, k, k/2)))/2)}
%Y Columns are: A000012, A001405(n-1), A047749 (k=3), A369930 (k=4), A143546 (k=5), A143547 (k=7), A143554 (k=9), A192893 (k=11).
%Y Cf. A070914, A303694, A303929, A361236, A361239, A370060, A370062.
%K nonn,tabl
%O 0,14
%A _Andrew Howroyd_, Feb 07 2024
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