A123611 Row sums of triangle A123610.

1, 2, 4, 8, 20, 52, 160, 492, 1620, 5408, 18504, 64132, 225440, 800048, 2865720, 10341208, 37568340, 137270956, 504176992, 1860277044, 6892335720, 25631327688, 95640894056, 357975249028, 1343650267296, 5056424257552

Offset: 0

Author

Paul D. Hanna, Oct 03 2006

Keywords

• nonn

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.

Crossrefs

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

Cf. A128545.

Programs

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 *)

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))))}

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