login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A054368 Number of unlabeled 6-gonal cacti having n polygons. 4
1, 1, 1, 4, 25, 187, 1772, 17880, 191967, 2141232, 24640989, 290610414, 3498042924, 42831369777, 532148952720, 6695274478834, 85166167050949, 1093843440166718, 14169564589464986, 184957445502335682, 2430876839834279341, 32147041999684759275, 427520786795342624432 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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

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 (pdf, dvi).

Index entries for sequences related to cacti

FORMULA

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

MAPLE

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

MATHEMATICA

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-Fran├žois Alcover, Jul 01 2018, after Andrew Howroyd *)

PROG

(PARI) 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

CROSSREFS

Column k=6 of A303694.

Cf. A054366, A054367.

Sequence in context: A051820 A246539 A166697 * A135147 A221382 A215791

Adjacent sequences:  A054365 A054366 A054367 * A054369 A054370 A054371

KEYWORD

nonn

AUTHOR

Simon Plouffe

EXTENSIONS

More terms from Zerinvary Lajos, Dec 01 2006

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

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 30 09:22 EDT 2020. Contains 333125 sequences. (Running on oeis4.)