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A369929
Array read by antidiagonals: T(n,k) is the number of achiral noncrossing partitions composed of n blocks of size k.
9
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
0,14
COMMENTS
T(n,2*k-1) is the number of achiral noncrossing k-gonal cacti with n polygons.
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.
FORMULA
T(n,k) = 2*A303929(n,k) - A303694(n,k).
T(n,2*k-1) = 2*A361239(n,k) - A361236(n,k).
EXAMPLE
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 ...
...
PROG
(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)}
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).
Sequence in context: A322763 A213211 A294775 * A330461 A332649 A321724
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Feb 07 2024
STATUS
approved