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A268919
Numerators of the rational number triangle R(n, k) = (n^4 - 30*n^2*k^2 + 60*n*k^3 -30*k^4) / (120*n), n >= 1, k = 1, ..., n. This is a regularized Sum_{j >= 0} (k + n*j)^(-s) for s = -3 defined by analytic continuation of a generalized Hurwitz zeta function.
5
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.
FORMULA
T(n, k) = numerator(R(n, k)) with the rational triangle R(n, k) = (n^4 - 30*n^2*k^2 + 60*n*k^3 - 30*k^4)/(120*n), n >= 1, k = 1, ..., n.
EXAMPLE
The triangle T(n, k) begins:
n\k 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
10: 757 29 -323 -91 -175 -91 -323 29 757 25
...
The triangle of the rationals R(n, k) begins:
n\k 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
7: 1321/840 -599/840 -1919/840 -1919/840 -599/840 1321/840 343/120;
8: 1313/480 -7/30 -1327/480 -56/15 -1327/480 -7/30 1313/480 64/15;
...
n=1, k=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 k^2+60m k^3-30k^4)/(120m), {m, 12}, {k, m}]]// Numerator (* Harvey P. Dale, Mar 03 2020 *)
PROG
(Magma)
A268919:= func< n, k | Numerator((n^4-30*n^2*k^2+60*n*k^3-30*k^4)/(120*n)) >;
[A268919(n, k): k in [1..n], n in [1..15]]; // G. C. Greubel, Oct 04 2024
(SageMath)
def A268919(n, k): return numerator((n^4-30*n^2*k^2+60*n*k^3-30*k^4)/(120*n))
flatten([[A268919(n, k) for k in range(1, n+1)] for n in range(1, 16)]) # G. C. Greubel, Oct 04 2024
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
KEYWORD
sign,frac,tabl,easy
AUTHOR
Wolfdieter Lang, Feb 25 2016
STATUS
approved