%I #17 Nov 22 2018 02:41:42

%S 0,0,1,1,1,5,5,7,14,3,15,55,11,13,91,35,20,34,51,57,95,35,77,253,46,

%T 25,325,117,63,203,145,155,248,44,187,595,105,111,703,247,130,205,287,

%U 301,473,165,345,1081,188,98,1225,425,221,689,477,495,770,133,551

%N Numerators of binomial(n-1, 2)/(6*n), for n >= 1. Numerators of Dedekind sum s(1, n).

%C For the denominators see A264389.

%C This gives the numerators of the rational numbers r(n) = s(1,n), where s(h,k) = Sum_{r=1..(k-1)} (r/k)*(h*r/k - floor(h*r/k)- 1/2), k >=1, are the Dedekind sums. See the Apostol reference, pp. 52, 61-69, 72-73, and the Weisstein link, where GCD(h,k) = 1 is assumed.

%C s(h,k) = Sum_{r = 1..k} ((r/k))*((h*r/k)) with ((x)) = x - floor(x) - 1/2 if x is not an integer, else 0.

%C s(h,k) = (Sum_{r=1..(k-1)} cot(Pi*h*r/k)*cot(Pi*r/k))/(4*k), k >= 1, r and h integers. Exercise 11, p. 72 of the Apostol reference.

%C 6*n*s(1,n) = binomial(n-1, 2) = A161680(n-1), n >= 1.

%D Apostol, Tom, M., Modular Functions and Dirichlet Series in Number Theory, Second edition, Springer, 1990.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DedekindSum.html">Dedekind Sum</a>.

%F a(n) = numerator(binomial(n-1, 2)/(6*n)) (in lowest terms), n >= 1.

%F a(n) = numerator(r(n)), with r(n) = s(1,n) = Sum_{r=1..(n-1)} (r/n)*(r/n - floor(r/n)- 1/2), n >= 1. For other forms see the above comments.

%o (Julia)

%o using Nemo

%o A264388(n) = numerator(dedekind_sum(1, n))

%o [A264388(n) for n in 1:70] |> println # _Peter Luschny_, Mar 13 2018

%Y Cf. A161680, A264389.

%K nonn,easy,frac

%O 1,6

%A _Wolfdieter Lang_, Jan 11 2016