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A268917
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Numerators of the rational number triangle R(m, a) = -a*(m-a)*(m - 2*a)/(6*m), m >= 1, a = 1, ..., m. This is a regularized Sum_{j >= 0} (a + m*j)^(-s) for s = -2 defined by analytic continuation of a generalized Hurwitz Zeta function.
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4
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0, 0, 0, -1, 1, 0, -1, 0, 1, 0, -2, -1, 1, 2, 0, -5, -4, 0, 4, 5, 0, -5, -5, -2, 2, 5, 5, 0, -7, -1, -5, 0, 5, 1, 7, 0, -28, -35, -1, -10, 10, 1, 35, 28, 0, -6, -8, -7, -4, 0, 4, 7, 8, 6, 0
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OFFSET
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1,11
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COMMENTS
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For the denominator triangle see A268918.
For details and the Hurwitz reference 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) = -a*(m - a)*(m - 2*a)/(6*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 ...
1: 0
2: 0 0
3: -1 1 0
4: -1 0 1 0
5: -2 -1 1 2 0
6: -5 -4 0 4 5 0
7: -5 -5 -2 2 5 5 0
8: -7 -1 -5 0 5 1 7 0
9: -28 -35 -1 -10 10 1 35 28 0
10: -6 -8 -7 -4 0 4 7 8 6 0
...
The triangle of the rationals R(m, a) begins:
m\a 1 2 3 4 5 6 7 8
1: 0/1
2: 0/1 0/1
3: -1/9 1/9 0/1
4: -1/4 0/1 1/4 0/1
5: -2/5 -1/5 1/5 2/5 0/1
6: -5/9 -4/9 0/1 4/9 5/9 0/1
7: -5/7 -5/7 -2/7 2/7 5/7 5/7 10/1
8: -7/8 -1/1 -5/8 0/1 5/8 1/1 7/8 0/1
...
Row m=9: -28/27 -35/27 -1/1 -10/27 10/27 1/1 35/27 28/27 0/1;
Row m=10:-6/5 -8/5 -7/5 -4/5 0/1 4/5 7/5 8/5 6/5 0/1.
...
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MATHEMATICA
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Numerator@ Table[-a (m - a) (m - 2 a)/(6 m), {m, 12}, {a, m}] // Flatten (* Michael De Vlieger, Feb 26 2016 *)
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PROG
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(PARI) tabl(nn) = {for (n=1, nn, for (k=1, n, print1(numerator(-k*(n-k)*(n-2*k)/(6*n)), ", "); ); print(); ); } \\ Michel Marcus, Feb 26 2016
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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