A268919 Numerators of the rational number triangle R(m, a) = (m^4 - 30*m^2*a^2 + 60*m*a^3 -30*a^4) / (120*m), m >= 1, a = 1, ..., m. This is a regularized Sum_{j >= 0} (a + m*j)^(-s) for s = -3 defined by analytic continuation of a generalized Hurwitz Zeta function.

1, -7, 1, -13, -13, 9, -7, -7, -7, 8, 29, -91, -91, 29, 25, 91, -13, -63, -13, 91, 9, 1321, -599, -1919, -1919, -599, 1321, 343, 1313, -7, -1327, -56, -1327, -7, 1313, 64, 1547, 227, -117, -1813, -1813, -117, 227, 1547, 243, 757, 29, -323, -91, -175, -91, -323, 29, 757, 25, 11641, 4921, -2639, -8879, -12359, -12359, -8879, -2639, 4921, 11641, 1331, 2851, 91, -63, -104, -2669, -63, -2669, -104, -63, 91, 2851, 72

Offset

1,2

Comments

For the denominator triangle see A268920.

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^4 - 30*m^2*a^2 + 60*m*a^3 - 30*a^4)/(120*m), m >= 1, a = 1, ..., m.

Example

The triangle T(m. a) begins:
m\a   1    2    3      4     5    6    7    8    9
1:    1
2:   -7    1
3:  -13  -13     9
4:   -7   -7    -7     8
5:   29  -91   -91    29    25
6:   91  -13   -63   -13    91    9
7: 1321 -599 -1919 -1919  -599 1321  343
8: 1313   -7 -1327   -56 -1327   -7 1313   64
9: 1547  227  -117 -1813 -1813 -117  227 1547  243
...
Row m=10: 757  29 -323   -91  -175  -91 -323 29 757  25;
...
The triangle of the rationals R(m, a) begins:
m\a   1        2        3      4       5      6
1:   1/120
2:  -7/120   1/15
3  -13/120 -13/120     9/40
4:  -7/240   -7/15   -7/240   8/15
5:  29/120 -91/120  -91/120  29/120   25/24
6:  91/120  -13/15   -63/40  -13/15  91/120  9/5
...
Row m=7: 1321/840 -599/840 -1919/840 -1919/840 -599/840 1321/840 343/120;
Row m=8: 1313/480  -7/30 -1327/480 -56/15 -1327/480 -7/30 1313/480  64/15;
Row m=9: 1547/360  227/360  -117/40 -1813/360 -1813/360 -117/40  227/360  1547/360 243/40;
Row m=10: 757/120  29/15 -323/120 -91/15 -175/24  -91/15 -323/120  29/15 757/120 25/3; ...
m=1, a=1: R(1, 1) = Sum_{j >= 1} j^3 = Zeta(-3) = -B_4/4 = -(-1/30)/4 = + 1/120, with the Bernoulli number B_4 = A027641(4)/A027642(4) = -1/30.

Mathematica

Flatten[Table[(m^4-30m^2 a^2+60m a^3-30a^4)/(120m), {m, 12}, {a, m}]]// Numerator (* Harvey P. Dale, Mar 03 2020 *)

Crossrefs

Cf. A268920 (denominators), A267863/A267864 (n=0), A268915/A268916 (n=1), A268917/A268918 (n=2).

Sequence in context: A060394 A203810 A225017 * A040055 A317016 A348983

Adjacent sequences: A268916 A268917 A268918 * A268920 A268921 A268922

Keyword

sign,frac,tabl,easy

Author

Wolfdieter Lang, Feb 25 2016

Status

approved