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 A268915 Numerators of the rational number triangle R(m, a) = - (m^2 - 6*m*a + 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. 5
 -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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,12 COMMENTS For the denominator triangle see A268916. For details and the Hurwitz reference see A267863. LINKS FORMULA T(m, a) = numerator(R(m, a)) with the rational triangle R(m, a) = - (m^2 - 6*m*a + a^2)/(12*m), m >= 1, a = 1, ..., m. EXAMPLE The triangle T(m. a) 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 ... Row n=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 ... 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 ... Row n=8: -11/48  1/12  13/48  1/3  13/48  1/12 -11/48  -2/3; Row n=9: -11/36  1/36  1/4 13/36 13/36 1/4  1/36   -11/36  -3/4; Row n=10: -23/60 -1/30  13/60  11/30  5/12 11/30  13/60  -1/30 -23/60  -5/6; Row m=11: -61/132  -13/132  23/132  47/132  59/132 59/132  47/132  23/132  -13/132  -61/132  -11/12; Row m=12: -13/24 -1/6  1/8  1/3  11/24  1/2  11/24 1/3  1/8 -1/6 -13/24  -1. ... 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 STATUS approved

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Last modified September 22 22:38 EDT 2021. Contains 347609 sequences. (Running on oeis4.)