OFFSET
0,4
COMMENTS
Also, the number of noncrossing partitions up to rotation composed of n blocks of size 3. - Andrew Howroyd, May 04 2018
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..200
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
Miklos Bona, Michel Bousquet, Gilbert Labelle, and Pierre Leroux, Enumeration of m-ary cacti, Advances in Applied Mathematics, 24 (2000), 22-56.
M. Bousquet and C. Lamathe, Enumeration of solid trees according to edge number and edge degree distribution, Discr. Math., 298 (2005), 115-141.
FORMULA
a(n) = ((Sum_{d|n} phi(n/d)*binomial(3*d, d)) + (Sum_{d|gcd(n-1, 3)} phi(d)*binomial(3*n/d, (n-1)/d)))/(3*n) - binomial(3*n, n)/(2*n+1) for n > 0. - Andrew Howroyd, May 04 2018
a(n) ~ 3^(3*n - 1/2) / (sqrt(Pi) * n^(5/2) * 2^(2*n + 2)). - Vaclav Kotesovec, Jun 01 2022
MAPLE
with(combinat): with(numtheory): m := 3: for p from 1 to 40 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: # James A. Sellers, Mar 17 2000
MATHEMATICA
a[0] = 1;
a[n_] := (DivisorSum[n, EulerPhi[n/#] Binomial[3 #, #]&] + DivisorSum[GCD[n - 1, 3], EulerPhi[#] Binomial[3n/#, (n-1)/#]&])/(3n) - Binomial[3n, n]/ (2n + 1);
Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Jul 02 2018, after Andrew Howroyd *)
PROG
(PARI) a(n) = {if(n==0, 1, (sumdiv(n, d, eulerphi(n/d)*binomial(3*d, d)) + sumdiv(gcd(n-1, 3), d, eulerphi(d)*binomial(3*n/d, (n-1)/d)))/(3*n) - binomial(3*n, n)/(2*n+1))} \\ Andrew Howroyd, May 04 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Simon Plouffe, Mar 15 2000
EXTENSIONS
More terms from James A. Sellers, Mar 17 2000
Terms a(24) and beyond from Andrew Howroyd, May 04 2018
STATUS
approved