A268917 - OEIS

%I #26 Oct 05 2024 09:40:20

%S 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,

%T -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,

%U -15,-21,-20,-14,-5,5,14,20,21,15,0,-55,-20,-9,-16,-35,0,35,16,9,20,55,0

%N Numerators of the rational number triangle R(n, k) = -k*(n-k)*(n - 2*k)/(6*n), n >= 1, k = 1, ..., n. This is a regularized Sum_{j >= 0} (k + n*j)^(-s) for s = -2 defined by analytic continuation of a generalized Hurwitz zeta function.

%C For the denominator triangle see A268918.

%C For details and the Hurwitz reference see A267863.

%H G. C. Greubel, <a href="/A268917/b268917.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) = -k*(n - k)*(n - 2*k)/(6*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  10 ...

%e   1:    0

%e   2:    0   0

%e   3:   -1   1   0

%e   4:   -1   0   1   0

%e   5:   -2  -1   1   2   0

%e   6:   -5  -4   0   4   5  0

%e   7:   -5  -5  -2   2   5  5   0

%e   8:   -7  -1  -5   0   5  1   7   0

%e   9:  -28 -35  -1 -10  10  1  35  28   0

%e   10:  -6  -8  -7  -4   0  4   7   8   6   0

%e   ...

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

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

%e   1:   0/1

%e   2:   0/1  0/1

%e   3:  -1/9  1/9   0/1

%e   4:  -1/4  0/1   1/4   0/1

%e   5:  -2/5 -1/5   1/5   2/5   0/1

%e   6:  -5/9 -4/9   0/1   4/9   5/9   0/1

%e   7:  -5/7 -5/7  -2/7   2/7   5/7   5/7  10/1

%e   8:  -7/8 -1/1  -5/8   0/1   5/8   1/1  7/8   0/1

%e   ...

%e Row m=9: -28/27 -35/27 -1/1 -10/27  10/27  1/1  35/27  28/27  0/1;

%e Row m=10:-6/5 -8/5 -7/5 -4/5  0/1  4/5  7/5 8/5    6/5  0/1.

%e ...

%t Numerator@ Table[-k (m - k) (m - 2 k)/(6 m), {m, 15}, {k, m}] // Flatten (* _Michael De Vlieger_, Feb 26 2016 *)

%o (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

%o (Magma)

%o A268917:= func< n,k | Numerator(k*(k-n)*(n-2*k)/(6*n)) >;

%o [A268917(n,k): k in [1..n], n in [1..15]]; // _G. C. Greubel_, Oct 04 2024

%o (SageMath)

%o def A268917(n,k): return numerator(k*(k-n)*(n-2*k)/(6*n))

%o flatten([[A268917(n,k) for k in range(1,n+1)] for n in range(1,16)]) # _G. C. Greubel_, Oct 04 2024

%Y Cf. A268918 (denominators), A267863/A267864 (n=0),  A268915/A268916 (n=1), A268919/A268920 (n=3).

%K sign,frac,tabl,easy

%O 1,11

%A _Wolfdieter Lang_, Feb 24 2016

%E More terms added by _G. C. Greubel_, Oct 04 2024