A054367 -id:A054367 - cp's OEIS frontend

A303913 Array read by antidiagonals: T(n,k) is the number of (planar) unlabeled asymmetric k-ary cacti having n polygons.

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

Offset: 0

Andrew Howroyd, May 02 2018

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

			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 ...
...

Andrew Howroyd, Table of n, a(n) for n = 0..1274
Miklos Bona, Michel Bousquet, Gilbert Labelle, Pierre Leroux, Enumeration of m-ary cacti, arXiv:math/9804119 [math.CO], 1998-1999.
Wikipedia, Cactus graph
Index entries for sequences related to cacti

Columns k=2..7 are A054358, A054422, A052395, A054364, A054367, A054370.

Cf. A303694, A303912.

T[0, _] = 1;
T[n_, k_] := DivisorSum[n, MoebiusMu[n/#] Binomial[k #, #] &]/n - (k-1) Binomial[n k, n]/((k-1) n + 1);
Table[T[n-k, k], {n, 0, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, May 22 2018 *)

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))}

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.

A054366 Number of unlabeled 6-ary cacti having n polygons.

Original entry on oeis.org

1, 1, 6, 21, 146, 1101, 10632, 107062, 1151802, 12845442, 147845706, 1743640908, 20988257544, 256987965379, 3192893716320, 40171643847696, 510997002280522, 6563060603543658, 85017387536789916, 1109744672540225367, 14585261039005676046

Offset: 0

Simon Plouffe

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..200
Miklos Bona, Michel Bousquet, Gilbert Labelle and Pierre Leroux, Enumeration of m-ary cacti, Advances in Applied Mathematics, 24 (2000), 22-56.
Index entries for sequences related to cacti

Column k=6 of A303912.

Cf. A054367, A054368.

a[n_] := If[n == 0, 1, (Binomial[6*n, n]/(5 n + 1) + DivisorSum[n, Binomial[6*#, #]*EulerPhi[n/#]*Boole[# < n] & ])/n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jul 17 2017 *)

a(n) = if(n==0, 1, sumdiv(n, d, eulerphi(n/d)*binomial(6*d, d))/n - 5*binomial(6*n, n)/(5*n+1)) \\ Andrew Howroyd, May 02 2018

a(n) = (1/n)*(Sum_{d|n} phi(n/d)*binomial(6*d, d)) - 5*binomial(6*n, n)/(5*n+1) for n > 0. - Andrew Howroyd, May 02 2018

a(n) ~ sqrt(3) * 6^(6*n) / (sqrt(Pi) * n^(5/2) * 5^(5*n + 3/2)). - Vaclav Kotesovec, Jul 17 2017

More terms from Jean-François Alcover, Jul 17 2017

A054368 Number of unlabeled 6-gonal cacti having n polygons.

Original entry on oeis.org

1, 1, 1, 4, 25, 187, 1772, 17880, 191967, 2141232, 24640989, 290610414, 3498042924, 42831369777, 532148952720, 6695274478834, 85166167050949, 1093843440166718, 14169564589464986, 184957445502335682, 2430876839834279341, 32147041999684759275, 427520786795342624432

Offset: 0

Simon Plouffe

nonn

Also, the number of noncrossing partitions up to rotation composed of n blocks of size 6. - Andrew Howroyd, May 04 2018

Andrew Howroyd, Table of n, a(n) for n = 0..200
Miklos Bona, Michel Bousquet, Gilbert Labelle, and Pierre Leroux, Enumeration of m-ary cacti, Advances in Applied Mathematics, 24 (2000), 22-56.
Index entries for sequences related to cacti

Column k=6 of A303694.

Cf. A054366, A054367.

with(combinat): with(numtheory): m := 6: for p from 2 to 28 do s1 := 0: s2 := 0: for d from 1 to p do if p mod d = 0 then s1 := s1+phi(p/d)*binomial(m*d, d) fi: od: for d from 1 to p-1 do if gcd(m, p-1) mod d = 0 then s2 := s2+phi(d)*binomial((p*m)/d, (p-1)/d) fi: od: printf(`%d, `, (s1+s2)/(m*p)-binomial(m*p, p)/(p*(m-1)+1)) od: # Zerinvary Lajos, Dec 01 2006

a[0] = 1;
a[n_] := (DivisorSum[n, EulerPhi[n/#] Binomial[6#, #]&] + DivisorSum[GCD[n - 1, 6], EulerPhi[#] Binomial[6n/#, (n-1)/#]&])/(6n) - Binomial[6n, n]/(5 n + 1);
Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Jul 01 2018, after Andrew Howroyd *)

a(n) = {if(n==0, 1, (sumdiv(n, d, eulerphi(n/d)*binomial(6*d, d)) + sumdiv(gcd(n-1, 6), d, eulerphi(d)*binomial(6*n/d, (n-1)/d)))/(6*n) - binomial(6*n, n)/(5*n+1))} \\ Andrew Howroyd, May 04 2018

a(n) = ((Sum_{d|n} phi(n/d)*binomial(6*d, d)) + (Sum_{d|gcd(n-1, 6)} phi(d)*binomial(6*n/d, (n-1)/d)))/(6*n) - binomial(6*n, n)/(5*n+1) for n > 0. - Andrew Howroyd, May 04 2018

More terms from Zerinvary Lajos, Dec 01 2006

Terms a(21) and beyond from Andrew Howroyd, May 04 2018

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A303913 Array read by antidiagonals: T(n,k) is the number of (planar) unlabeled asymmetric k-ary cacti having n polygons.

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A054366 Number of unlabeled 6-ary cacti having n polygons.

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A054368 Number of unlabeled 6-gonal cacti having n polygons.

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