cp's OEIS Frontend

A361239 Array read by antidiagonals: T(n,k) is the number of noncrossing k-gonal cacti with n polygons up to rotation and reflection.

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