A268917 Numerators of the rational number triangle R(n, k) = -k*(n-k)*(n - 2*k)/(6*n), n >= 1, k = 1, ..., n. This is a regularized Sum_{j >= 0} (k + n*j)^(-s) for s = -2 defined by analytic continuation of a generalized Hurwitz zeta function.

0, 0, 0, -1, 1, 0, -1, 0, 1, 0, -2, -1, 1, 2, 0, -5, -4, 0, 4, 5, 0, -5, -5, -2, 2, 5, 5, 0, -7, -1, -5, 0, 5, 1, 7, 0, -28, -35, -1, -10, 10, 1, 35, 28, 0, -6, -8, -7, -4, 0, 4, 7, 8, 6, 0, -15, -21, -20, -14, -5, 5, 14, 20, 21, 15, 0, -55, -20, -9, -16, -35, 0, 35, 16, 9, 20, 55, 0

Offset

1,11

Comments

For the denominator triangle see A268918.

For details and the Hurwitz reference see A267863.

Links

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Formula

T(n, k) = numerator(R(n, k)) with the rational triangle R(n, k) = -k*(n - k)*(n - 2*k)/(6*n), n >= 1, k = 1, ..., n.

Example

The triangle T(n, k) begins:
  n\k   1   2   3   4   5  6   7   8   9  10 ...
  1:    0
  2:    0   0
  3:   -1   1   0
  4:   -1   0   1   0
  5:   -2  -1   1   2   0
  6:   -5  -4   0   4   5  0
  7:   -5  -5  -2   2   5  5   0
  8:   -7  -1  -5   0   5  1   7   0
  9:  -28 -35  -1 -10  10  1  35  28   0
  10:  -6  -8  -7  -4   0  4   7   8   6   0
  ...
The triangle of the rationals R(n, k) begins:
  n\k   1    2     3     4     5     6    7     8
  1:   0/1
  2:   0/1  0/1
  3:  -1/9  1/9   0/1
  4:  -1/4  0/1   1/4   0/1
  5:  -2/5 -1/5   1/5   2/5   0/1
  6:  -5/9 -4/9   0/1   4/9   5/9   0/1
  7:  -5/7 -5/7  -2/7   2/7   5/7   5/7  10/1
  8:  -7/8 -1/1  -5/8   0/1   5/8   1/1  7/8   0/1
  ...
Row m=9: -28/27 -35/27 -1/1 -10/27  10/27  1/1  35/27  28/27  0/1;
Row m=10:-6/5 -8/5 -7/5 -4/5  0/1  4/5  7/5 8/5    6/5  0/1.
...

Mathematica

Numerator@ Table[-k (m - k) (m - 2 k)/(6 m), {m, 15}, {k, m}] // Flatten (* Michael De Vlieger, Feb 26 2016 *)

Prog

(PARI) tabl(nn) = {for (n=1, nn, for (k=1, n, print1(numerator(-k*(n-k)*(n-2*k)/(6*n)), ", "); ); print(); ); } \\ Michel Marcus, Feb 26 2016
(Magma)
A268917:= func< n, k | Numerator(k*(k-n)*(n-2*k)/(6*n)) >;
[A268917(n, k): k in [1..n], n in [1..15]]; // G. C. Greubel, Oct 04 2024
(SageMath)
def A268917(n, k): return numerator(k*(k-n)*(n-2*k)/(6*n))
flatten([[A268917(n, k) for k in range(1, n+1)] for n in range(1, 16)]) # G. C. Greubel, Oct 04 2024

Crossrefs

Cf. A268918 (denominators), A267863/A267864 (n=0), A268915/A268916 (n=1), A268919/A268920 (n=3).

Sequence in context: A244658 A117586 A307988 * A176811 A057594 A259029

Adjacent sequences: A268914 A268915 A268916 * A268918 A268919 A268920

Keyword

sign,frac,tabl,easy

Author

Wolfdieter Lang, Feb 24 2016

Extensions

More terms added by G. C. Greubel, Oct 04 2024

Status

approved