OFFSET
2,11
COMMENTS
For the denominators see A278716.
The Dedekind sums are s(h,k) = Sum_{r=1..k-1} (r/k)*(h*r/k - floor(h*r/k) - 1/2) = Sum_{r=1..k-1} ((r/k))*((h*r)/k) with the period 1 sawtooth function ((x)) = x - floor(x) - 1/2 if x is not an integer and 0 otherwise. One assumes gcd(h,k) = 1, and for other h, k values the table has 0's. See the references Apostol pp. 52, 61-69, 72-73 and Ayoub pp. 168, 191. See also the Weisstein link.
In order to have a regular triangle T(k, h) one starts with k >= 2 and h = 1, 2, ..., k-1. s(1,1) = 0.
REFERENCES
Apostol, Tom, M., Modular Functions and Dirichlet Series in Number Theory, Second edition, Springer, 1990.
Ayoub, R., An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963.
LINKS
G. C. Greubel, Rows n=2..100 of triangle, flattened
Eric Weisstein's World of Mathematics, Dedekind Sum.
FORMULA
T(k ,h) = numerator(s(h,k)) with the Dedekind sums s(h,k) given in a comment above and gcd(h,k) = 1. k >=2, h = 1, 2, ..., k-1. If gcd(h,k) is not 1 then T(k,h) is put to 0 (in the example o is used). Note that T(k,h) can vanish also for gcd(h,k) = 1.
EXAMPLE
The triangle T(k,h) begins (if gcd(k,h) is not 1 we use o instead of 0):
k\h 1 2 3 4 5 6 7 8 9 10 11 12
2: 0
3: 1 -1
4: 1 o -1
5: 1 0 0 -1
6: 5 o o o -5
7: 5 1 -1 1 -1 -5
8: 7 o 1 o -1 o -7
9: 14 4 o -4 4 o -4 -14
10: 3 o 0 o o o 0 o -3
11: 15 5 3 3 -5 5 -3 -3 -5 -15
12: 55 o o o -1 o 1 o o o -55
13: 11 4 1 -1 0 -4 4 0 1 -1 -4 -11
...
n = 14: 13 o 3 o 3 o o o -3 o -3 o -13,
n = 15: 91 7 0 19 0 0 -7 7 0 0 -19 0 -7 -91.
...
---------------------------------------------
The rational triangle s(h,k) begins (here o is used if gcd(h,k) is not 1):
k\h 1 2 3 4 5 6 7
2: 0
3: 1/18 -1/18
4: 1/8 o -1/8
5: 1/5 0 0 -1/5
6: 5/18 o o o -5/18
7: 5/14 1/14 -1/14 1/14 -1/14 -5/14
8: 7/16 o 1/16 o -1/16 o -7/16
...
n = 9: 14/27 4/27 o -4/27 4/27 o -4/27 -14/27,
n = 10: 3/5 o 0 o o o 0 o -3/5,
n = 11: 15/22 5/22 3/22 3/22 -5/22 5/22 -3/22 -3/22 -5/22 -15/22,
n = 12: 55/72 o o o -1/72 o 1/72 o o o -55/72,
n = 13: 11/13 4/13 1/13 -1/13 0 -4/13 4/13 0 1/13 -1/13 -4/13 -11/13,
n = 14: 13/14 o 3/14 o 3/14 o o o -3/14 o -3/14 o -13/14,
n = 15: 1/90 7/18 o 19/90 o o -7/18 7/18 o o -19/90 o -7/18 -91/90.
...
--------------------------------------------
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]; Numerator[Table[T[n, k], {n, 2, 15}, {k, 1, n-1}]] //Flatten (* G. C. Greubel, Nov 22 2018 *)
PROG
(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(numerator(T(n, k)), ", "))) \\ G. C. Greubel, Nov 22 2018
(Magma) [[GCD(n, k) eq 1 select Numerator((&+[(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..15]]; // G. C. Greubel, Nov 22 2018
(Sage)
def T(n, k):
if gcd(n, k)==1:
return numerator(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
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Nov 28 2016
STATUS
approved