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A054362 Number of unlabeled 4-gonal cacti having n polygons. 5
1, 1, 1, 3, 11, 52, 307, 1936, 13207, 93496, 683988, 5127163, 39230669, 305299420, 2410624122, 19273255184, 155780437711, 1271253542364, 10462650241996, 86765190816362, 724450039738076, 6086167189623746, 51416796881915019 (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 4. - Andrew Howroyd, Apr 30 2018

LINKS

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

Miklos Bona, Michel Bousquet, Gilbert Labelle, 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(4*d, d)) + (Sum_{d|gcd(n-1, 4)} phi(d)*binomial(4*n/d, (n-1)/d)))/(4*n) - binomial(4*n, n)/(3*n+1) for n > 0. - Andrew Howroyd, Apr 30 2018

MAPLE

with(combinat): with(numtheory): m := 4: 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[4#, #]&] + DivisorSum[GCD[n - 1, 4], EulerPhi[#] Binomial[4n/#, (n-1)/#]&])/(4n) - Binomial[4n, n]/ (3n + 1);

Table[a[n], {n, 0, 22}] (* Jean-Fran├žois Alcover, Jun 29 2018, after Andrew Howroyd *)

PROG

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

CROSSREFS

Column k=4 of A303694.

Cf. A052394, A052395, A303870.

Sequence in context: A179322 A014510 A058799 * A129833 A321585 A107958

Adjacent sequences:  A054359 A054360 A054361 * A054363 A054364 A054365

KEYWORD

nonn

AUTHOR

Simon Plouffe

EXTENSIONS

More terms from Zerinvary Lajos, Dec 01 2006

STATUS

approved

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Last modified March 1 03:32 EST 2021. Contains 341732 sequences. (Running on oeis4.)