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A361239
Array read by antidiagonals: T(n,k) is the number of noncrossing k-gonal cacti with n polygons up to rotation and reflection.
6
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, 124, 108, 1, 1, 1, 1, 9, 57, 349, 931, 507, 1, 1, 1, 1, 10, 85, 737, 3766, 7801, 2431, 1, 1, 1, 1, 12, 117, 1359, 10601, 45632, 68685, 12441, 1
OFFSET
0,14
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals).
Wikipedia, Cactus graph.
FORMULA
T(0,k) = T(1,k) = T(2,k) = 1.
T(2*n,k) = (A361236(2*n,k) + binomial((2*k-1)*n + 1, n)/((2*k-1)*n + 1))/2.
T(2*n+1,k) = (A361236(2*n+1,k) + k*binomial((2*k-1)*n + k, n)/((2*k-1)*n + k))/2.
EXAMPLE
Array begins:
===================================================
n\k | 1 2 3 4 5 6 ...
----+----------------------------------------------
0 | 1 1 1 1 1 1 ...
1 | 1 1 1 1 1 1 ...
2 | 1 1 1 1 1 1 ...
3 | 1 3 4 6 7 9 ...
4 | 1 7 19 35 57 85 ...
5 | 1 28 124 349 737 1359 ...
6 | 1 108 931 3766 10601 24112 ...
7 | 1 507 7801 45632 167741 471253 ...
8 | 1 2431 68685 580203 2790873 9678999 ...
9 | 1 12441 630850 7687128 48300850 206780448 ...
...
PROG
(PARI) \\ R(n, k) gives A361236.
u(n, k, r) = {r*binomial(n*(2*k-1) + r, n)/(n*(2*k-1) + r)}
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)))}
T(n, k) = {(R(n, k) + u(n\2, k, if(n%2, k, 1)))/2}
CROSSREFS
Columns 1..4 are A000012, A296533, A361240, A361241.
Row n=3 is A032766.
Sequence in context: A046534 A224489 A318933 * A140334 A131324 A371884
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Mar 06 2023
STATUS
approved