%I #29 Jun 11 2024 23:46:57

%S 1,2,4,8,20,52,160,492,1620,5408,18504,64132,225440,800048,2865720,

%T 10341208,37568340,137270956,504176992,1860277044,6892335720,

%U 25631327688,95640894056,357975249028,1343650267296,5056424257552

%N Row sums of triangle A123610.

%H Andrew Howroyd, <a href="/A123611/b123611.txt">Table of n, a(n) for n = 0..200</a>

%H Michal Bassan, Serte Donderwinkel, and Brett Kolesnik, <a href="https://arxiv.org/abs/2406.05110">Graphical sequences and plane trees</a>, arXiv:2406.05110 [math.CO], 2024.

%F 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.

%F 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.

%F a(n) = (1/n) * Sum_{d | n} phi(n/d) * binomial(2*d, d) for n>0. - _Andrew Howroyd_, Apr 02 2017

%F 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

%t 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 *)

%o (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))))}

%Y Cf. A047996, A003239; A123610 (triangle), A123612 (antidiagonal sums); central terms: A123617, A123618.

%Y Cf. A128545.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Oct 03 2006