OFFSET
0,2
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..200
Michal Bassan, Serte Donderwinkel, and Brett Kolesnik, Graphical sequences and plane trees, arXiv:2406.05110 [math.CO], 2024.
FORMULA
a(n) = 2*A047996(2*n,n) = 2*A003239(n) for n > 0, where A047996 is the triangle of circular binomial coefficients and A003239(n) = number of rooted planar trees with n non-root nodes.
Also equals the row sums of triangle A128545, where A128545(n,k) is the coefficient of q^(n*k) in the q-binomial coefficient [2n,n] for n >= k >= 0.
a(n) = (1/n) * Sum_{d | n} phi(n/d) * binomial(2*d, d) for n>0. - Andrew Howroyd, Apr 02 2017
G.f.: 1 - Sum_{n>=1} (phi(n)/n) * log((1-2*x^n + sqrt(1-4*x^n))/2) = 1 - 2*Sum_{n>=1} (phi(n)/n) * log((1+sqrt(1-4*x^n))/2). (Except for the term a(0) = 1, the first g.f. follows from the g.f. in A123610 by setting y=1, as suggested by P. D. Hanna.) - Petros Hadjicostas, Oct 24 2017
MATHEMATICA
Total /@ Table[If[k == 0, 1, 1/n DivisorSum[n, If[Divisible[k, #], EulerPhi[#] Binomial[n/#, k/#]^2, 0] &]], {n, 0, 25}, {k, 0, n}] (* Michael De Vlieger, Apr 03 2017, after Jean-François Alcover at A123610 *)
PROG
(PARI) {a(n)=sum(k=0, n, if(k==0, 1, (1/n)*sumdiv(n, d, if(gcd(k, d)==d, eulerphi(d)*binomial(n/d, k/d)^2, 0))))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 03 2006
STATUS
approved