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A268915
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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.
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5
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-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
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OFFSET
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1,12
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COMMENTS
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For the denominator triangle see A268916.
For details and the Hurwitz reference (f(-1, a) on page 92) see A267863.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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
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
...
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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