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

1,6

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

Table of n, a(n) for n=1..59.

Eric Weisstein's World of Mathematics, Dedekind Sum.

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.

Prog

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

Crossrefs

Cf. A161680, A264389.

Sequence in context: A252006 A158299 A093307 * A141392 A088047 A109257

Adjacent sequences: A264385 A264386 A264387 * A264389 A264390 A264391

Keyword

nonn,easy,frac

Author

Wolfdieter Lang, Jan 11 2016

Status

approved