A268915 Numerators of the rational number triangle R(m, a) = - (m^2 - 6*m*a + 6*a^2)/(12*m), m >= 1, a = 1, ..., m. This is a regularized Sum_{j >= 0} (a + m*j) defined by analytic continuation of a generalized Hurwitz zeta function.

-1, 1, -1, 1, 1, -1, 1, 1, 1, -1, -1, 11, 11, -1, -5, -1, 1, 1, 1, -1, -1, -13, 11, 23, 23, 11, -13, -7, -11, 1, 13, 1, 13, 1, -11, -2, -11, 1, 1, 13, 13, 1, 1, -11, -3, -23, -1, 13, 11, 5, 11, 13, -1, -23, -5, -61, -13, 23, 47, 59, 59, 47, 23, -13, -61, -11, -13, -1, 1, 1, 11, 1, 11, 1, 1, -1, -13, -1

Offset

1,12

Comments

For the denominator triangle see A268916.

For details and the Hurwitz reference (f(-1, a) on page 92) 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) = - (n^2 - 6*n*k + 6*k^2)/(12*n), n >= 1, k = 1, ..., n.

Example

The triangle T(m, k) begins:
  m\a   1   2   3   4   5   6   7   8   9  10  11
  1:   -1
  2:    1  -1
  3:    1   1  -1
  4:    1   1   1  -1
  5:   -1  11  11  -1  -5
  6:   -1   1   1   1  -1  -1
  7:  -13  11  23  23  11 -13  -7
  8:  -11   1  13   1  13   1 -11  -2
  9:  -11   1   1  13  13   1   1 -11  -3
  10: -23  -1  13  11   5  11  13  -1 -23  -5
  11: -61 -13  23  47  59  59  47  23 -13 -61 -11
  12: -13 -1  1  1  11  1  11  1  1  -1  -13  -1.
  ...
The triangle of rationals R(m, a) begins:
  m\a   1     2     3     4     5      6      7      8      9    10 ...
  1:  -1/12
  2:   1/12  -1/6
  3:   1/12  1/12  -1/4
  4:   1/24   1/6  1/24  -1/3
  5:  -1/60 11/60 11/60 -1/60 -5/12
  6:  -1/12   1/6   1/4   1/6 -1/12   -1/2
  7: -13/84 11/84 23/84 23/84 11/84 -13/84  -7/12
  8: -11/48  1/12 13/48   1/3 13/48   1/12 -11/48   -2/3
  9: -11/36  1/36   1/4 13/36 13/36    1/4   1/36 -11/36   -3/4
  10: 23/60 -1/30 13/60 11/30  5/12  11/30  13/60  -1/30 -23/60 -5/6
  ...

Mathematica

A268915[n_, k_]:= Numerator[-(n^2 -6*n*k +6*k^2)/(12*n)];
Table[A268915[n, k], {n, 15}, {k, n}]//Flatten (* G. C. Greubel, Oct 04 2024 *)

Prog

(Magma)
A268915:= func< n, k |  Numerator(-(n^2 - 6*n*k + 6*k^2)/(12*n)) >;
[A268915(n, k): k in [1..n], n in [1..15]]; // G. C. Greubel, Oct 04 2024
(SageMath)
def A2686915(n, k): return numerator(-(n^2 -6*n*k +6*k^2)/(12*n))
flatten([[A2686915(n, k) for k in range(1, n+1)] for n in range(1, 16)]) # G. C. Greubel, Oct 04 2024

Crossrefs

Cf. A268915 (denominators), A267863/A267864 (n=0), A268917/A268918 (n=2), A268919/A268920 (n=3).

Sequence in context: A262014 A287288 A287979 * A332731 A322270 A260589

Adjacent sequences: A268912 A268913 A268914 * A268916 A268917 A268918

Keyword

sign,frac,tabl,easy

Author

Wolfdieter Lang, Feb 18 2016

Extensions

Definition corrected by Georg Fischer, Mar 15 2022

Status

approved