A278716 Triangle read by rows: T(k, h) gives the denominators of the Dedekind sums s(h, k) with gcd(h, k) = 1 or 0 otherwise.

1, 18, 18, 8, 1, 8, 5, 1, 1, 5, 18, 1, 1, 1, 18, 14, 14, 14, 14, 14, 14, 16, 1, 16, 1, 16, 1, 16, 27, 27, 1, 27, 27, 1, 27, 27, 5, 1, 1, 1, 1, 1, 1, 1, 5, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 72, 1, 1, 1, 72, 1, 72, 1, 1, 1, 72, 13, 13, 13, 13, 1, 13, 13, 1, 13, 13, 13, 13

Offset: 2

Author

Wolfdieter Lang, Nov 28 2016

Keywords

• nonn
• tabl
• frac
• easy

Comments

For the numerators see A278715, also for references and details of the Dedekind sums s(h, k).

Examples

			The triangle T(k, h) begins (here l is used if gcd(h, k) > 1 instead of 1):
k\h  1  2  3  4   5  6   7   8  9 10 11 12
2:   1
3:  18 18
4:   8  l  8
5:   5  1  1  5
6:  18  l  l  l  18
7:  14 14 14 14  14  14
8:  16  l 16  l  16  l  16
9:  27 27  l 27  27  l  27  27
10:  5  l  1  l   l  l   1   l  5
11: 22 22 22 22  22 22  22  22 22 22
12: 72  l  l  l  72  l  72   l  l  l 72
13: 13 13 13 13   1 13  13   1 13 13 13 13
...
n = 14: 14 l 14 l 14 l l l 14 l 14 l 14,
n = 15: 90 18 l 90 l l 18 18 l l 90 l 18 90.
...
		

Links

• G. C. Greubel, Rows n=2..100 of triangle, flattened

Crossrefs

Cf. A278715.

Programs

Magma

[[GCD(n,k) eq 1 select Denominator((&+[(j/n)*(k*j/n - Floor(k*j/n) - 1/2): j in [1..(n-1)]])) else 0: k in [1..(n-1)]]: n in [2..10]]; // G. C. Greubel, Nov 22 2018

Mathematica

T[n_, k_]:= If[GCD[n, k] == 1, Sum[(j/n)*(k*j/n - Floor[k*j/n] - 1/2), {j, 1, n - 1}], 0]; Denominator[Table[T[n, k], {n, 2, 15}, {k, 1, n - 1}]]//Flatten (* G. C. Greubel, Nov 22 2018 *)

PARI

{T(n,k) = if(gcd(n,k)==1, sum(j=1,n-1, (j/n)*(k*j/n - floor(k*j/n) - 1/2)), 0)};
for(n=2,15, for(k=1,n-1, print1(denominator(T(n,k)), ", "))) \\ G. C. Greubel, Nov 22 2018

Sage

def T(n,k):
    if gcd(n,k)==1:
       return denominator(sum((j/n)*(k*j/n - floor(k*j/n) - 1/2) for j in (1..(n-1))))
    elif gcd(n,k)!=1:
        return 0
    else:
        0
[[T(n,k) for k in (1..n-1)] for n in (2..15)] # G. C. Greubel, Nov 22 2018

Formula

T(k ,h) = denominator(s(h,k)) with the Dedekind sums s(h,k) given in a comment on A278715 and gcd(h,k) = 1. k >= 2, h = 1, 2, ..., k-1. If gcd(h,k) > 1 then T(h, k) = 1 (from s(h,k) put to 0).