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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

Table of n, a(n) for n=1..78.

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.)