%I #21 Sep 08 2022 08:46:15
%S -1,0,-1,1,-1,-1,1,0,-1,-1,3,1,-1,-3,-1,1,1,0,-1,-1,-1,5,3,1,-1,-3,-5,
%T -1,3,1,1,0,-1,-1,-3,-1,7,5,1,1,-1,-1,-5,-7,-1,2,3,1,1,0,-1,-1,-3,-2,
%U -1,9,7,5,3,1,-1,-3,-5,-7,-9,-1,5,1,1,1,1,0,-1,-1,-1,-1,-5,-1
%N Numerators of the rational number triangle R(m, a) = (m - 2*a)/(2*m), m >= 1, a = 1, ..., m. This is a regularized Sum_{j >= 0} (a + m*j)^(-s) for s = 0 defined by analytic continuation of a generalized Hurwitz Zeta function.
%C For the denominator triangle see A267864.
%C Hurwitz' generalized m-family of Zeta functions f(m; s, a) = Sum_{j = 0} 1/(a + m*j)^s, with m >= 1, a = 1, 2, ... , m, defined for Re(s) > 1, is analytically continued via a certain integral, and is then valid for complex s. See the Hurwitz reference where f(m; s, a) is called f(s, a), or f(s, a | m) if the m dependence is indicated. Here R(m, a) = f(m; 0, a).
%C The usual Hurwitz Zeta function is f(1; s, a). The usual Zeta function is f(1; s, 1).
%C Remember the Zeta function regularized Sum_{j >= 1} 1 = Zeta(0) = -1/2, (here a = m = 1). More generally, Zeta(1 - k) = -(-1)^k*B_k/k , k >= 1, with the Bernoulli numbers A027641/A027642.
%C The general analytically continued result of Hurwitz is f(m; -n, a) = ((-1)^n)*n!*[x^n] (exp((m-a)*x))/(exp(m*x) - 1) , n >= 0. (n is called rho in Hurwitz's article). Here n = 0.
%C For n = 1, 2, and 3 see A268915/A268916, A268917/A268918 and A268919/A268920, respectively.
%H G. C. Greubel, <a href="/A267863/b267863.txt">Rows n=1..100 of triangle, flattened</a>
%H A. Hurwitz, <a href="https://gdz.sub.uni-goettingen.de/id/PPN599415665_0027?tify={%22pages%22:[96,97]}">Eigenschaften der Dirichlet'schen Funktionen F(s) = Sum (D/n) 1/n^s, die bei der Bestimmung der Klassenzahlen binärer quadratischen Formen auftreten</a>, Z. f. Mathematik und Physik, 27 (1882) 86 - 101; Mathematische Werke, Band I (Funktionentheorie), 72 - 88.
%F T(m, a) = numerator(R(m, a)) with R(m, a) = (m-2*a)/(2*m), m >= 1, a = 1..m.
%e The triangle T(m, a) begins:
%e m\a 1 2 3 4 5 6 7 8 9 10 ...
%e 1: -1
%e 2: 0 -1
%e 3: 1 -1 -1
%e 4: 1 0 -1 -1
%e 5: 3 1 -1 -3 -1
%e 6: 1 1 0 -1 -1 -1
%e 7: 5 3 1 -1 -3 -5 -1
%e 8: 3 1 1 0 -1 -1 -3 -1
%e 9: 7 5 1 1 -1 -1 -5 -7 -1
%e 10: 2 3 1 1 0 -1 -1 -3 -2 -1
%e ...
%e The rational triangle R(m, a) begins:
%e m\a 1 2 3 4 5 6 7 8
%e 1: -1/2
%e 2: 0/1 -1/2
%e 3: 1/6 -1/6 -1/2
%e 4: 1/4 0/1 -1/4 -1/2
%e 5: 3/10 1/10 -1/10 -3/10 -1/2
%e 6: 1/3 1/6 0/1 -1/6 -1/3 -1/2
%e 7: 5/14 3/14 1/14 -1/14 -3/14 -5/14 -1/2
%e 8: 3/8 1/4 1/8 0/1 -1/8 -1/4 -3/8 -1/2
%e ...
%e Row n=9: 7/18 5/18 1/6 1/18 -1/18 -1/6 -5/18 -7/18 -1/2;
%e Row n=10: 2/5 3/10 1/5 1/10 0/1 -1/10 -1/5 -3/10 -2/5 -1/2.
%e ...
%t R[m_, a_] := HurwitzZeta[0, a/m]; (* or *) R[m_, a_] := (m - 2*a)/(2*m); Table[R[m, a] // Numerator, {m, 1, 12}, {a, 1, m}] // Flatten (* _Jean-François Alcover_, Feb 26 2016 *)
%o (PARI) T(m,n)=numerator(1/2 - n/m)
%o for(m=1,20, for(n=1,m, print1(T(m,n)", "))) \\ _Charles R Greathouse IV_, Jan 14 2017
%o (Magma) [[Numerator((n-2*k)/(2*n)): k in [1..n]]: n in [1..12]]; // _G. C. Greubel_, Feb 14 2019
%o (Sage) [[numerator((n-2*k)/(2*n)) for k in (1..n)] for n in (1..12)] # _G. C. Greubel_, Feb 14 2019
%Y Cf. A267864 (denominators), A268915/A268916 (n=1), A268917/A268918 (n=2), A268919/A268920 (n=3).
%K sign,frac,tabl,easy
%O 1,11
%A _Wolfdieter Lang_, Feb 18 2016