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%I #13 Mar 11 2023 00:13:49
%S 1,1,1,1,1,1,1,1,1,1,1,1,1,3,1,1,1,1,4,7,1,1,1,1,6,19,28,1,1,1,1,7,35,
%T 124,108,1,1,1,1,9,57,349,931,507,1,1,1,1,10,85,737,3766,7801,2431,1,
%U 1,1,1,12,117,1359,10601,45632,68685,12441,1
%N Array read by antidiagonals: T(n,k) is the number of noncrossing k-gonal cacti with n polygons up to rotation and reflection.
%H Andrew Howroyd, <a href="/A361239/b361239.txt">Table of n, a(n) for n = 0..1325</a> (first 51 antidiagonals).
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Cactus_graph">Cactus graph</a>.
%H <a href="/index/Ca#cacti">Index entries for sequences related to cacti</a>.
%F T(0,k) = T(1,k) = T(2,k) = 1.
%F T(2*n,k) = (A361236(2*n,k) + binomial((2*k-1)*n + 1, n)/((2*k-1)*n + 1))/2.
%F T(2*n+1,k) = (A361236(2*n+1,k) + k*binomial((2*k-1)*n + k, n)/((2*k-1)*n + k))/2.
%e Array begins:
%e ===================================================
%e n\k | 1 2 3 4 5 6 ...
%e ----+----------------------------------------------
%e 0 | 1 1 1 1 1 1 ...
%e 1 | 1 1 1 1 1 1 ...
%e 2 | 1 1 1 1 1 1 ...
%e 3 | 1 3 4 6 7 9 ...
%e 4 | 1 7 19 35 57 85 ...
%e 5 | 1 28 124 349 737 1359 ...
%e 6 | 1 108 931 3766 10601 24112 ...
%e 7 | 1 507 7801 45632 167741 471253 ...
%e 8 | 1 2431 68685 580203 2790873 9678999 ...
%e 9 | 1 12441 630850 7687128 48300850 206780448 ...
%e ...
%o (PARI) \\ R(n,k) gives A361236.
%o u(n,k,r) = {r*binomial(n*(2*k-1) + r, n)/(n*(2*k-1) + r)}
%o R(n,k) = {if(n==0, 1, u(n, k, 1)/((k-1)*n+1) + sumdiv(gcd(k,n-1), d, if(d>1, eulerphi(d)*u((n-1)/d, k, 2*k/d)/k)))}
%o T(n, k) = {(R(n, k) + u(n\2, k, if(n%2, k, 1)))/2}
%Y Columns 1..4 are A000012, A296533, A361240, A361241.
%Y Row n=3 is A032766.
%Y Cf. A361236, A361243.
%K nonn,tabl
%O 0,14
%A _Andrew Howroyd_, Mar 06 2023
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