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A303913
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Array read by antidiagonals: T(n,k) is the number of (planar) unlabeled asymmetric k-ary cacti having n polygons.
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8
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1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 3, 2, 0, 1, 1, 0, 6, 10, 8, 0, 1, 1, 0, 10, 28, 54, 18, 0, 1, 1, 0, 15, 60, 193, 222, 61, 0, 1, 1, 0, 21, 110, 505, 1140, 1107, 170, 0, 1, 1, 0, 28, 182, 1095, 3876, 7688, 5346, 538, 0, 1, 1, 0, 36, 280, 2093, 10326, 33125, 52364, 27399, 1654, 0
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OFFSET
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0,19
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COMMENTS
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A k-ary cactus is a planar k-gonal cactus with vertices on each polygon numbered 1..k counterclockwise with shared vertices having the same number. In total there are always exactly k ways to number a given cactus since all polygons are connected. See the reference for a precise definition. - Andrew Howroyd, Feb 18 2020
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LINKS
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Miklos Bona, Michel Bousquet, Gilbert Labelle, Pierre Leroux, Enumeration of m-ary cacti, arXiv:math/9804119 [math.CO], 1998-1999.
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FORMULA
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T(n,k) = (Sum_{d|n} mu(n/d)*binomial(k*d, d))/n - (k-1)*binomial(k*n, n)/((k-1)*n+1)) for n > 0.
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EXAMPLE
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Array begins:
===============================================================
n\k| 1 2 3 4 5 6 7 8
---+-----------------------------------------------------------
0 | 1 1 1 1 1 1 1 1 ...
1 | 1 1 1 1 1 1 1 1 ...
2 | 0 0 0 0 0 0 0 0 ...
3 | 0 1 3 6 10 15 21 28 ...
4 | 0 2 10 28 60 110 182 280 ...
5 | 0 8 54 193 505 1095 2093 3654 ...
6 | 0 18 222 1140 3876 10326 23394 47208 ...
7 | 0 61 1107 7688 33125 107056 285383 662620 ...
8 | 0 170 5346 52364 290700 1149126 3621150 9702008 ...
9 | 0 538 27399 373560 2661100 12845166 47813367 147765409 ...
...
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MATHEMATICA
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T[0, _] = 1;
T[n_, k_] := DivisorSum[n, MoebiusMu[n/#] Binomial[k #, #] &]/n - (k-1) Binomial[n k, n]/((k-1) n + 1);
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PROG
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(PARI) T(n, k)={if(n==0, 1, sumdiv(n, d, moebius(n/d)*binomial(k*d, d))/n - (k-1)*binomial(k*n, n)/((k-1)*n+1))}
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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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