A264388 Numerators of binomial(n-1, 2)/(6*n), for n >= 1. Numerators of Dedekind sum s(1, n).

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, 25, 325, 117, 63, 203, 145, 155, 248, 44, 187, 595, 105, 111, 703, 247, 130, 205, 287, 301, 473, 165, 345, 1081, 188, 98, 1225, 425, 221, 689, 477, 495, 770, 133, 551

Offset: 1

Author

Wolfdieter Lang, Jan 11 2016

Keywords

• nonn
• easy
• frac

Comments

For the denominators see A264389.

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.

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.

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.

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

References

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

Links

• Eric Weisstein's World of Mathematics, Dedekind Sum.

Crossrefs

Cf. A161680, A264389.

Programs

Julia

using Nemo
A264388(n) = numerator(dedekind_sum(1, n))
[A264388(n) for n in 1:70] |> println # Peter Luschny, Mar 13 2018

Formula

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

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.