A268919 - OEIS

%I #18 Oct 05 2024 09:40:11

%S 1,-7,1,-13,-13,9,-7,-7,-7,8,29,-91,-91,29,25,91,-13,-63,-13,91,9,

%T 1321,-599,-1919,-1919,-599,1321,343,1313,-7,-1327,-56,-1327,-7,1313,

%U 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

%N 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.

%C For the denominator triangle see A268920.

%C For details and the Hurwitz reference see A267863.

%H G. C. Greubel, <a href="/A268919/b268919.txt">Rows n = 1..50 of the triangle, flattened</a>

%F 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.

%e The triangle T(n, k) begins:

%e   n\k   1    2    3      4     5    6    7    8    9

%e   1:    1

%e   2:   -7    1

%e   3:  -13  -13     9

%e   4:   -7   -7    -7     8

%e   5:   29  -91   -91    29    25

%e   6:   91  -13   -63   -13    91    9

%e   7: 1321 -599 -1919 -1919  -599 1321  343

%e   8: 1313   -7 -1327   -56 -1327   -7 1313   64

%e   9: 1547  227  -117 -1813 -1813 -117  227 1547  243

%e   10: 757   29  -323   -91  -175  -91 -323   29  757  25

%e   ...

%e The triangle of the rationals R(n, k) begins:

%e   n\k   1        2        3      4       5      6

%e   1:    1/120

%e   2:   -7/120     1/15

%e   3   -13/120  -13/120      9/40

%e   4:   -7/240    -7/15    -7/240     8/15

%e   5:   29/120  -91/120   -91/120    29/120    25/24

%e   6:   91/120   -13/15    -63/40   -13/15     91/120    9/5

%e   7: 1321/840 -599/840 -1919/840 -1919/840  -599/840 1321/840  343/120;

%e   8: 1313/480    -7/30 -1327/480   -56/15  -1327/480   -7/30  1313/480   64/15;

%e   ...

%e 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.

%t 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 *)

%o (Magma)

%o A268919:= func< n,k | Numerator((n^4-30*n^2*k^2+60*n*k^3-30*k^4)/(120*n)) >;

%o [A268919(n,k): k in [1..n], n in [1..15]]; // _G. C. Greubel_, Oct 04 2024

%o (SageMath)

%o def A268919(n,k): return numerator((n^4-30*n^2*k^2+60*n*k^3-30*k^4)/(120*n))

%o flatten([[A268919(n,k) for k in range(1,n+1)] for n in range(1,16)]) # _G. C. Greubel_, Oct 04 2024

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

%K sign,frac,tabl,easy

%O 1,2

%A _Wolfdieter Lang_, Feb 25 2016