A267863 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.

-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, -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, -1, 9, 7, 5, 3, 1, -1, -3, -5, -7, -9, -1, 5, 1, 1, 1, 1, 0, -1, -1, -1, -1, -5, -1

Offset

1,11

Comments

For the denominator triangle see A267864.

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).

The usual Hurwitz Zeta function is f(1; s, a). The usual Zeta function is f(1; s, 1).

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.

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.

For n = 1, 2, and 3 see A268915/A268916, A268917/A268918 and A268919/A268920, respectively.

Links

G. C. Greubel, Rows n=1..100 of triangle, flattened

A. Hurwitz, Eigenschaften der Dirichlet'schen Funktionen F(s) = Sum (D/n) 1/n^s, die bei der Bestimmung der Klassenzahlen binärer quadratischen Formen auftreten, Z. f. Mathematik und Physik, 27 (1882) 86 - 101; Mathematische Werke, Band I (Funktionentheorie), 72 - 88.

Formula

T(m, a) = numerator(R(m, a)) with R(m, a) = (m-2*a)/(2*m), m >= 1, a = 1..m.

Example

The triangle T(m, a) begins:
m\a   1  2  3  4  5  6  7   8  9  10 ...
1:   -1
2:    0 -1
3:    1 -1 -1
4:    1  0 -1 -1
5:    3  1 -1 -3 -1
6:    1  1  0 -1 -1 -1
7:    5  3  1 -1 -3 -5 -1
8:    3  1  1  0 -1 -1 -3 -1
9:    7  5  1  1 -1 -1 -5 -7 -1
10:   2  3  1  1  0 -1 -1 -3 -2  -1
...
The rational triangle R(m, a) begins:
m\a  1     2     3     4     5     6     7    8
1: -1/2
2:  0/1  -1/2
3:  1/6  -1/6  -1/2
4:  1/4   0/1  -1/4  -1/2
5:  3/10  1/10 -1/10 -3/10 -1/2
6:  1/3   1/6   0/1  -1/6  -1/3  -1/2
7:  5/14  3/14  1/14 -1/14 -3/14 -5/14 -1/2
8:  3/8   1/4   1/8   0/1  -1/8  -1/4  -3/8  -1/2
...
Row n=9: 7/18  5/18  1/6   1/18 -1/18 -1/6  -5/18 -7/18 -1/2;
Row n=10: 2/5   3/10  1/5   1/10  0/1  -1/10 -1/5  -3/10 -2/5 -1/2.
...

Mathematica

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 *)

Prog

(PARI) T(m, n)=numerator(1/2 - n/m)
for(m=1, 20, for(n=1, m, print1(T(m, n)", "))) \\ Charles R Greathouse IV, Jan 14 2017
(Magma) [[Numerator((n-2*k)/(2*n)): k in [1..n]]: n in [1..12]]; // G. C. Greubel, Feb 14 2019
(Sage) [[numerator((n-2*k)/(2*n)) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Feb 14 2019

Crossrefs

Cf. A267864 (denominators), A268915/A268916 (n=1), A268917/A268918 (n=2), A268919/A268920 (n=3).

Sequence in context: A322506 A103496 A345441 * A262681 A076498 A110268

Adjacent sequences: A267860 A267861 A267862 * A267864 A267865 A267866

Keyword

sign,frac,tabl,easy

Author

Wolfdieter Lang, Feb 18 2016

Status

approved