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%I #16 Mar 15 2022 18:22:23
%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.
%F T(m, a) = numerator(R(m, a)) with the rational triangle R(m, a) = - (m^2 - 6*m*a + 6*a^2)/(12*m), m >= 1, a = 1, ..., m.
%e The triangle T(m. a) 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 ...
%Y Cf. A268915 (denominators), A267863/A267864 (n=0), A268917/A268918 (n=2), A268919/A268920 (n=3).
%K sign,frac,tabl,easy
%O 1,12
%A _Wolfdieter Lang_, Feb 18 2016
%E Definition corrected by _Georg Fischer_, Mar 15 2022
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