A303913 -id:A303913 - cp's OEIS frontend

A303912 Array read by antidiagonals: T(n,k) is the number of (planar) unlabeled k-ary cacti having n polygons.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 4, 6, 6, 1, 1, 1, 5, 10, 19, 10, 1, 1, 1, 6, 15, 44, 57, 28, 1, 1, 1, 7, 21, 85, 197, 258, 63, 1, 1, 1, 8, 28, 146, 510, 1228, 1110, 190, 1, 1, 1, 9, 36, 231, 1101, 4051, 7692, 5475, 546, 1, 1, 1, 10, 45, 344, 2100, 10632, 33130, 52828, 27429, 1708, 1

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.

			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  | 1   2     3      4       5        6        7         8 ...
3  | 1   3     6     10      15       21       28        36 ...
4  | 1   6    19     44      85      146      231       344 ...
5  | 1  10    57    197     510     1101     2100      3662 ...
6  | 1  28   258   1228    4051    10632    23884     47944 ...
7  | 1  63  1110   7692   33130   107062   285390    662628 ...
8  | 1 190  5475  52828  291925  1151802  3626295   9711032 ...
9  | 1 546 27429 373636 2661255 12845442 47813815 147766089 ...
...

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 2..7 are A054357, A052393, A052394, A054363, A054366, A054369.

Cf. A070914, A303694, A303913.

T[0, _] = 1;
T[n_, k_] := DivisorSum[n, EulerPhi[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, eulerphi(n/d)*binomial(k*d, d))/n - (k-1)*binomial(k*n, n)/((k-1)*n+1))}

T(n,k) = (Sum_{d|n} phi(n/d)*binomial(k*d, d))/n - (k-1)*binomial(k*n, n)/((k-1)*n+1) for n > 0.

T(n,k) ~ A070914(n,k-1)/n for fixed k > 1.

A052395 Number of unlabeled asymmetric 4-ary cacti having n polygons.

Original entry on oeis.org

1, 1, 0, 6, 28, 193, 1140, 7688, 52364, 373560, 2732836, 20506254, 156899748, 1221179922, 9642327324, 77092881840, 623120435820, 5085013101160, 41850590485164, 347060754685884, 2897800074184240, 24344668688255109, 205667186830447412, 1746375819789491992

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=4 of A303913.

Cf. A052394, A054362.

a[0] = 1;
a[n_] := DivisorSum[n, MoebiusMu[n/#] Binomial[4#, #]&]/n - 3 Binomial[4n, n]/(3n + 1);
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Jun 29 2018, after Andrew Howroyd *)

a(n) = if(n==0, 1, sumdiv(n, d, moebius(n/d)*binomial(4*d, d))/n - 3*binomial(4*n, n)/(3*n+1)) \\ Andrew Howroyd, Apr 30 2018

a(n) = (1/n)*(Sum_{d|n} mu(n/d)*binomial(4*d, d)) - 3*binomial(4*n, n)/(3*n+1) for n > 0. - Andrew Howroyd, Apr 30 2018

Terms a(13) and beyond from Andrew Howroyd, May 02 2018

A054364 Number of unlabeled asymmetric 5-ary cacti having n polygons.

Original entry on oeis.org

1, 1, 0, 10, 60, 505, 3876, 33125, 290700, 2661100, 25049020, 241724375, 2379812100, 23833198135, 242172147380, 2491817140380, 25921361665100, 272256630756260, 2884054853862540, 30784716141936520, 330853931834416520, 3577823885432126890, 38907658110093347780

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=5 of A303913.

Cf. A054363, A054365.

a[0] = 1;
a[n_] := DivisorSum[n, MoebiusMu[n/#] Binomial[5#, #]&]/n - 4 Binomial[5n, n]/(4n+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, moebius(n/d)*binomial(5*d, d))/n - 4*binomial(5*n, n)/(4*n+1)) \\ Andrew Howroyd, May 02 2018

a(n) = (1/n)*(Sum_{d|n} mu(n/d)*binomial(5*d, d)) - 4*binomial(5*n, n)/(4*n+1) for n > 0. - Andrew Howroyd, May 02 2018

Terms a(13) and beyond from Andrew Howroyd, May 02 2018

A054367 Number of unlabeled asymmetric 6-ary cacti having n polygons.

Original entry on oeis.org

1, 1, 0, 15, 110, 1095, 10326, 107056, 1149126, 12845166, 147817170, 1743640902, 20987930112, 256987965373, 3192889862238, 40171643818920, 510996955110550, 6563060603543652, 85017386945583408, 1109744672540225361, 14585261031466256370, 192882251992018723947

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 A303913.

Cf. A054366, A054368.

a[0] = 1;
a[n_] := DivisorSum[n, MoebiusMu[n/#] Binomial[6#, #]&]/n - 5 Binomial[6n, n]/(5n + 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, moebius(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} mu(n/d)*binomial(6*d, d)) - 5*binomial(6*n, n)/(5*n+1) for n > 0. - Andrew Howroyd, May 02 2018

Terms a(11) and beyond from Andrew Howroyd, May 02 2018

A054370 Number of unlabeled asymmetric 7-ary cacti having n polygons.

Original entry on oeis.org

1, 1, 0, 21, 182, 2093, 23394, 285383, 3621150, 47813367, 650302814, 9066061193, 128986881660, 1866877313441, 27417577343674, 407771633368995, 6131640430397790, 93096368350684720, 1425633583562073228, 21998953427963954547, 341803226976420690600

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=7 of A303913.

Cf. A054369, A054371.

a[0] = 1;
a[n_] := DivisorSum[n, MoebiusMu[n/#] Binomial[7#, #]&]/n - 6 Binomial[7n, n]/(6n + 1);
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jul 01 2018, after Andrew Howroyd *)

a(n) = if(n==0, 1, sumdiv(n, d, moebius(n/d)*binomial(7*d, d))/n - 6*binomial(7*n, n)/(6*n+1)) \\ Andrew Howroyd, May 02 2018

a(n) = (1/n)*(Sum_{d|n} mu(n/d)*binomial(7*d, d)) - 6*binomial(7*n, n)/(6*n+1) for n > 0. - Andrew Howroyd, May 02 2018

Terms a(11) and beyond from Andrew Howroyd, May 02 2018

A054422 Number of unlabeled asymmetric ternary cacti having n triangles.

Original entry on oeis.org

1, 1, 0, 3, 10, 54, 222, 1107, 5346, 27399, 142770, 764967, 4170672, 23140813, 130189302, 741650172, 4270501218, 24825326196, 145534796520, 859627488963, 5112003992610, 30586307195304, 184023393204654, 1112800162657899

Offset: 0

Simon Plouffe, Mar 15 2000

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=3 of A303913.

Cf. A052393, A054423.

a[0] = 1; a[n_] := (1/n) Sum[MoebiusMu[n/d] Binomial[3d, d], {d, Divisors[n] } ] - 2 Binomial[3n, n]/(2n + 1);
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Jul 24 2018, after Andrew Howroyd *)

a(n) = if(n==0, 1, sumdiv(n, d, moebius(n/d)*binomial(3*d, d))/n - 2*binomial(3*n, n)/(2*n+1)) \\ Andrew Howroyd, May 02 2018

a(n) = (1/n)*(Sum_{d|n} mu(n/d)*binomial(3*d, d)) - 2*binomial(3*n, n)/(2*n+1) for n > 0. - Andrew Howroyd, May 02 2018

More terms from James Sellers, Mar 16 2000

A054358 Number of unlabeled asymmetric 2-ary cacti having n polygons.

Original entry on oeis.org

1, 1, 0, 1, 2, 8, 18, 61, 170, 538, 1654, 5344, 17252, 57146, 190786, 646305, 2209050, 7626164, 26532732, 93013852, 328196780, 1165060170, 4158266282, 14915635376, 53745892932, 194477856048, 706436256598, 2575316698792, 9419568272632

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=2 of A303913.

Cf. A002995, A054357.

a[0] = 1; a[n_] := DivisorSum[n, MoebiusMu[n/#] Binomial[2#, #]&]/n - Binomial[2n, n]/(n+1);
Table[a[n], {n, 0, 28}] (* Jean-François Alcover, Jul 01 2018, after Andrew Howroyd *)

a(n) = if(n==0, 1, sumdiv(n, d, moebius(n/d)*binomial(2*d, d))/n - binomial(2*n, n)/(n+1)) \\ Andrew Howroyd, May 02 2018

a(n) = (1/n)*(Sum_{d|n} mu(n/d)*binomial(2*d, d)) - binomial(2*n, n)/(n+1) for n > 0. - Andrew Howroyd, May 02 2018

More terms from Vladeta Jovovic, Oct 04 2007

cp's OEIS Frontend

A303912 Array read by antidiagonals: T(n,k) is the number of (planar) unlabeled k-ary cacti having n polygons.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A052395 Number of unlabeled asymmetric 4-ary cacti having n polygons.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A054364 Number of unlabeled asymmetric 5-ary cacti having n polygons.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A054367 Number of unlabeled asymmetric 6-ary cacti having n polygons.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A054370 Number of unlabeled asymmetric 7-ary cacti having n polygons.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A054422 Number of unlabeled asymmetric ternary cacti having n triangles.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A054358 Number of unlabeled asymmetric 2-ary cacti having n polygons.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions