A054365 -id:A054365

Search: a054365 -id:a054365

Displaying 1-4 of 4 results found.

page 1

A303694

Array read by antidiagonals: T(n,k) is the number of noncrossing partitions up to rotation composed of n blocks of size k.

+10
15

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, 7, 6, 1, 1, 1, 1, 3, 11, 19, 14, 1, 1, 1, 1, 4, 17, 52, 86, 34, 1, 1, 1, 1, 4, 25, 102, 307, 372, 95, 1, 1, 1, 1, 5, 33, 187, 811, 1936, 1825, 280, 1, 1, 1, 1, 5, 43, 300, 1772, 6626, 13207, 9143, 854, 1

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,14

COMMENTS

Also, the number of unlabeled planar k-gonal cacti having n polygons.

The number of noncrossing partitions counted distinctly is given by A070914(n,k-1).

LINKS

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], Apr 1998.

Wikipedia, Cactus graph

Wikipedia, Noncrossing partition

Index entries for sequences related to cacti

FORMULA

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

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

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 7 11 17 25 33 43 55 ...

5 | 1 6 19 52 102 187 300 463 663 ...

6 | 1 14 86 307 811 1772 3412 5993 9821 ...

7 | 1 34 372 1936 6626 17880 40770 82887 154079 ...

8 | 1 95 1825 13207 58385 191967 518043 1213879 2558305 ...

9 | 1 280 9143 93496 532251 2141232 6830545 18471584 44121134 ...

...

MATHEMATICA

T[0, _] = 1;

T[n_, k_] := (DivisorSum[n, EulerPhi[n/#] Binomial[k #, #]&] + DivisorSum[ GCD[n-1, k], EulerPhi[#] Binomial[n k/#, (n-1)/#]&])/(k n) - Binomial[k n, n]/(n (k-1) + 1);

Table[T[n-k, k], {n, 0, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, May 22 2018 *)

PROG

(PARI) T(n, k)={if(n==0, 1, (sumdiv(n, d, eulerphi(n/d)*binomial(k*d, d)) + sumdiv(gcd(n-1, k), d, eulerphi(d)*binomial(n*k/d, (n-1)/d)))/(k*n) - binomial(k*n, n)/(n*(k-1)+1))}

CROSSREFS

Columns 2..7 are A002995(n+1), A054423, A054362, A054365, A054368, A054371.

Cf. A070914, A209805, A303864, A303929.

KEYWORD

nonn,tabl

AUTHOR

Andrew Howroyd, Apr 28 2018

STATUS

approved

A054363

Number of unlabeled 5-ary cacti having n polygons.

+10
4

1, 1, 5, 15, 85, 510, 4051, 33130, 291925, 2661255, 25059670, 241724380, 2379912355, 23833198140, 242173108050, 2491817151160, 25921371278805, 272256630756265, 2884054952424115, 30784716141936525, 330853932861650870, 3577823885433087690, 38907658120970944700

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

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

FORMULA

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

a(n) ~ 5^(5*n + 1/2) / (sqrt(Pi) * n^(5/2) * 2^(8*n + 7/2)). - Vaclav Kotesovec, Jul 17 2017

MATHEMATICA

a[n_] := If[n == 0, 1, (Binomial[5*n, n]/(4*n + 1) + DivisorSum[n, Binomial[5*#, #]*EulerPhi[n/#]*Boole[# < n] & ])/n]; Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Jul 17 2017 *)

PROG

(PARI) a(n) = if(n==0, 1, sumdiv(n, d, eulerphi(n/d)*binomial(5*d, d))/n - 4*binomial(5*n, n)/(4*n+1)) \\ Andrew Howroyd, May 02 2018

CROSSREFS

Column k=5 of A303912.

Cf. A054364, A054365.

KEYWORD

nonn

AUTHOR

Simon Plouffe

EXTENSIONS

More terms from Jean-François Alcover, Jul 17 2017

STATUS

approved

A054364

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

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

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

FORMULA

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

MATHEMATICA

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 *)

PROG

(PARI) 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

CROSSREFS

Column k=5 of A303913.

Cf. A054363, A054365.

KEYWORD

nonn

AUTHOR

Simon Plouffe

EXTENSIONS

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

STATUS

approved

A303871

Number of noncrossing partitions up to rotation and reflection composed of n blocks of size 5.

+10
3

1, 1, 1, 3, 11, 60, 423, 3381, 29335, 266703, 2507232, 24177705, 238003111, 2383370158, 24217426745, 249182213284, 2592138293117, 27225668134063, 288405507217589, 3078471666603235, 33085393411436772, 357782389095170193, 3890765813426578535, 42527471172438573757

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..200

FORMULA

a(n) ~ 5^(5*n - 1/2) / (sqrt(Pi) * n^(5/2) * 2^(8*n + 9/2)). - Vaclav Kotesovec, Jun 01 2022

MATHEMATICA

u[n_, k_, r_] := (r*Binomial[k*n + r, n]/(k*n + r));

e[n_, k_] := Sum[ u[j, k, 1 + (n - 2*j)*k/2], {j, 0, n/2}]

c[n_, k_] := If[n == 0, 1, (DivisorSum[n, EulerPhi[n/#]*Binomial[k*#, #] &] + DivisorSum[GCD[n - 1, k], EulerPhi[#]*Binomial[n*k/#, (n - 1)/#] &])/(k*n) - Binomial[k*n, n]/(n*(k - 1) + 1)];

T[n_, k_] := (1/2)*(c[n, k] + If[n == 0, 1, If[OddQ[k], If[OddQ[n], 2*u[Quotient[n, 2], k, (k + 1)/2], u[n/2, k, 1] + u[n/2 - 1, k, k]], e[n, k] + If[OddQ[n], u[Quotient[n, 2], k, k/2]]]/2]) /. Null -> 0;

a[n_] := T[n, 5];

Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 14 2018, after Andrew Howroyd and A303929 *)

CROSSREFS

Column k=5 of A303929.

Cf. A054365.

KEYWORD

nonn

AUTHOR

Andrew Howroyd, May 01 2018

STATUS

approved

page 1

Search completed in 0.007 seconds