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A369929
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Array read by antidiagonals: T(n,k) is the number of achiral noncrossing partitions composed of n blocks of size k.
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9
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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
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OFFSET
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0,14
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COMMENTS
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T(n,2*k-1) is the number of achiral noncrossing k-gonal cacti with n polygons.
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LINKS
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FORMULA
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EXAMPLE
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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 ...
...
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PROG
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(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)}
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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