%I #28 Oct 05 2024 09:40:15

%S -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,

%T 11,-13,-7,-11,1,13,1,13,1,-11,-2,-11,1,1,13,13,1,1,-11,-3,-23,-1,13,

%U 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

%N 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.

%C For the denominator triangle see A268916.

%C For details and the Hurwitz reference (f(-1, a) on page 92) see A267863.

%H G. C. Greubel, <a href="/A268915/b268915.txt">Rows n = 1..50 of the triangle, flattened</a>

%F 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.

%e The triangle T(m, k) begins:

%e m\a 1 2 3 4 5 6 7 8 9 10 11

%e 1: -1

%e 2: 1 -1

%e 3: 1 1 -1

%e 4: 1 1 1 -1

%e 5: -1 11 11 -1 -5

%e 6: -1 1 1 1 -1 -1

%e 7: -13 11 23 23 11 -13 -7

%e 8: -11 1 13 1 13 1 -11 -2

%e 9: -11 1 1 13 13 1 1 -11 -3

%e 10: -23 -1 13 11 5 11 13 -1 -23 -5

%e 11: -61 -13 23 47 59 59 47 23 -13 -61 -11

%e 12: -13 -1 1 1 11 1 11 1 1 -1 -13 -1.

%e ...

%e The triangle of rationals R(m, a) begins:

%e m\a 1 2 3 4 5 6 7 8 9 10 ...

%e 1: -1/12

%e 2: 1/12 -1/6

%e 3: 1/12 1/12 -1/4

%e 4: 1/24 1/6 1/24 -1/3

%e 5: -1/60 11/60 11/60 -1/60 -5/12

%e 6: -1/12 1/6 1/4 1/6 -1/12 -1/2

%e 7: -13/84 11/84 23/84 23/84 11/84 -13/84 -7/12

%e 8: -11/48 1/12 13/48 1/3 13/48 1/12 -11/48 -2/3

%e 9: -11/36 1/36 1/4 13/36 13/36 1/4 1/36 -11/36 -3/4

%e 10: 23/60 -1/30 13/60 11/30 5/12 11/30 13/60 -1/30 -23/60 -5/6

%e ...

%t A268915[n_, k_]:= Numerator[-(n^2 -6*n*k +6*k^2)/(12*n)];

%t Table[A268915[n,k], {n,15}, {k,n}]//Flatten (* _G. C. Greubel_, Oct 04 2024 *)

%o (Magma)

%o A268915:= func< n,k | Numerator(-(n^2 - 6*n*k + 6*k^2)/(12*n)) >;

%o [A268915(n,k): k in [1..n], n in [1..15]]; // _G. C. Greubel_, Oct 04 2024

%o (SageMath)

%o def A2686915(n,k): return numerator(-(n^2 -6*n*k +6*k^2)/(12*n))

%o flatten([[A2686915(n,k) for k in range(1,n+1)] for n in range(1,16)]) # _G. C. Greubel_, Oct 04 2024

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

%K sign,frac,tabl,easy,changed

%O 1,12

%A _Wolfdieter Lang_, Feb 18 2016

%E Definition corrected by _Georg Fischer_, Mar 15 2022