This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A361236 #16 Mar 11 2023 00:14:05
%S A361236 1,1,1,1,1,1,1,1,1,1,1,1,1,4,1,1,1,1,5,11,1,1,1,1,8,33,49,1,1,1,1,9,
%T A361236 63,230,204,1,1,1,1,12,105,664,1827,984,1,1,1,1,13,159,1419,7462,
%U A361236 15466,4807,1,1,1,1,16,221,2637,21085,90896,137085,24739,1
%N A361236 Array read by antidiagonals: T(n,k) is the number of noncrossing k-gonal cacti with n polygons up to rotation.
%C A361236 The number of noncrossing k-gonal cacti is given by column 2*(k-1) of A070914. This sequence enumerates these cacti with rotations being considered equivalent.
%C A361236 Equivalently, T(n,k) is the number of connected acyclic k-uniform noncrossing antichains with n blocks covering (k-1)*n+1 nodes where the intersection of two blocks is at most 1 node modulo cyclic rotation of the nodes.
%C A361236 Noncrossing trees correspond to the case of k = 2.
%H A361236 Andrew Howroyd, <a href="/A361236/b361236.txt">Table of n, a(n) for n = 0..1325</a> (first 51 antidiagonals).
%H A361236 Wikipedia, <a href="https://en.wikipedia.org/wiki/Cactus_graph">Cactus graph</a>.
%H A361236 <a href="/index/Ca#cacti">Index entries for sequences related to cacti</a>.
%F A361236 T(0,k) = T(1,k) = T(2,k) = 1.
%e A361236 =====================================================
%e A361236 n\k | 1 2 3 4 5 6 ...
%e A361236 ----+------------------------------------------------
%e A361236 0 | 1 1 1 1 1 1 ...
%e A361236 1 | 1 1 1 1 1 1 ...
%e A361236 2 | 1 1 1 1 1 1 ...
%e A361236 3 | 1 4 5 8 9 12 ...
%e A361236 4 | 1 11 33 63 105 159 ...
%e A361236 5 | 1 49 230 664 1419 2637 ...
%e A361236 6 | 1 204 1827 7462 21085 48048 ...
%e A361236 7 | 1 984 15466 90896 334707 941100 ...
%e A361236 8 | 1 4807 137085 1159587 5579961 19354687 ...
%e A361236 9 | 1 24739 1260545 15369761 96589350 413533260 ...
%e A361236 ...
%o A361236 (PARI) \\ here u is Fuss-Catalan sequence with p = 2*k-1.
%o A361236 u(n,k,r) = {r*binomial(n*(2*k-1) + r, n)/(n*(2*k-1) + r)}
%o A361236 T(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)))
%Y A361236 Columns k=1..4 are A000012, A296532, A361237, A361238.
%Y A361236 Row n=3 is A042948.
%Y A361236 Cf. A070914, A303694, A303912, A361239, A361242.
%K A361236 nonn,tabl
%O A361236 0,14
%A A361236 _Andrew Howroyd_, Mar 05 2023