A369929 Array read by antidiagonals: T(n,k) is the number of achiral noncrossing partitions composed of n blocks of size k.

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, 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, 102, 55, 70, 1, 1, 1, 1, 5, 9, 34, 64, 136, 213, 143, 126, 1

Offset: 0

Author

Andrew Howroyd, Feb 07 2024

Keywords

• nonn
• tabl

Comments

T(n,2*k-1) is the number of achiral noncrossing k-gonal cacti with n polygons.

Examples

			Array begins:
===============================================
n\k| 1  2   3   4    5    6    7    8     9 ...
---+-------------------------------------------
0  | 1  1   1   1    1    1    1    1     1 ...
1  | 1  1   1   1    1    1    1    1     1 ...
2  | 1  1   1   1    1    1    1    1     1 ...
3  | 1  2   2   3    3    4    4    5     5 ...
4  | 1  3   3   5    5    7    7    9     9 ...
5  | 1  6   7  16   18   31   34   51    55 ...
6  | 1 10  12  31   35   64   70  109   117 ...
7  | 1 20  30 102  136  296  368  651   775 ...
8  | 1 35  55 213  285  663  819 1513  1785 ...
9  | 1 70 143 712 1155 3142 4495 9304 12350 ...
...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)
• Michel Bousquet and Cédric Lamathe, On symmetric structures of order two, Discrete Math. Theor. Comput. Sci. 10 (2008), 153-176. See Table 1.
• Wikipedia, Fuss-Catalan number.
• Wikipedia, Noncrossing partition.

Crossrefs

Columns are: A000012, A001405(n-1), A047749 (k=3), A369930 (k=4), A143546 (k=5), A143547 (k=7), A143554 (k=9), A192893 (k=11).

Cf. A070914, A303694, A303929, A361236, A361239, A370060, A370062.

Programs

PARI

\\ u(n,k,r) are Fuss-Catalan numbers.
u(n,k,r) = {r*binomial(k*n + r, n)/(k*n + r)}
e(n,k) = {sum(j=0, n\2, u(j, k, 1+(n-2*j)*k/2))}
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)}

Formula

T(n,k) = 2*A303929(n,k) - A303694(n,k).

T(n,2*k-1) = 2*A361239(n,k) - A361236(n,k).