A117972 Numerator of zeta'(-2n), n >= 0.

1, -1, 3, -45, 315, -14175, 467775, -42567525, 638512875, -97692469875, 9280784638125, -2143861251406875, 147926426347074375, -48076088562799171875, 9086380738369043484375, -3952575621190533915703125

Offset: 0

Author

Eric W. Weisstein, Apr 06 2006

Keywords

• sign
• frac

Comments

In A160464 the coefficients of the ES1 matrix are defined. This matrix led to the discovery that the successive differences of the ES1[1-2*m,n] coefficients for m = 1, 2, 3, ..., are equal to the values of zeta'(-2n), see also A094665 and A160468. - Johannes W. Meijer, May 24 2009

A048896(n), n >= 1: Numerators of Maclaurin series for 1 - ((sin x)/x)^2,

a(n), n >= 2: Denominators of Maclaurin series for 1 - ((sin x)/x)^2, the correlation function in Montgomery's pair correlation conjecture. - Daniel Forgues, Oct 16 2011

From Andrey Zabolotskiy, Sep 23 2021: (Start)

zeta'(-2n), which is mentioned in the Name, is irrational. For n > 0, a(n) is the numerator of the rational fraction g(n) = Pi^(2n)*zeta'(-2n)/zeta(2n+1). The denominator is 4*A048896(n-1). g(n) = f(n) for n > 0, where f(n) is given in the Formula section. Also, f(n) = Bernoulli(2n)/z(n)/4 (see Formula section) for all n.

For n = 0, zeta'(0) = -log(2Pi)/2, g(0) can be set to 0 because of the infinite denominator. However, a(0) is set to 1 because it is the numerator of f(0).

It seems that -4*f(n)*alpha_n = A000182(n), where alpha_n = A191657(n, p(n)) / A191658(n, p(n)) [where p(n) = A000041(n)] is the n-th "elementary coefficient" from the paper by Izaurieta et al. (End)

Examples

			-1/4, 3/4, -45/8, 315/4, -14175/8, 467775/8, -42567525/16, ...
-zeta(3)/(4*Pi^2), (3*zeta(5))/(4*Pi^4), (-45*zeta(7))/(8*Pi^6), (315*zeta(9))/(4*Pi^8), (-14175*zeta(11))/(8*Pi^10), ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Fernando Izaurieta, Ricardo Ramírez and Eduardo Rodríguez, Dirac Matrices for Chern-Simons Gravity, arXiv:1106.1648 [math-ph], 2011-2012.
• J. Sondow and E. W. Weisstein, MathWorld: Riemann Zeta Function

Crossrefs

Cf. A000367, A002445, A049606, A046988, A002432, A117973, A048896.

From Johannes W. Meijer, May 24 2009: (Start)

Cf. A160464, A094665 and A160468.

Absolute values equal row sums of A160468. (End)

Cf. A191657, A191658, A000182, A000041.

Programs

Maple

# Without rational arithmetic
a := n -> (-1)^n*(2*n)!*2^(add(i,i=convert(n,base,2))-2*n);
# Peter Luschny, May 02 2009

Mathematica

Table[Numerator[(2 n)!/2^(2 n + 1) (-1)^n], {n, 0, 30}]

Maxima

L:taylor(1/x*sin(sqrt(x))^2,x,0,15); makelist(denom(coeff(L,x,n))*(-1)^(n+1),n,0,15); /* Vladimir Kruchinin, May 30 2011 */

Formula

a(n) = numerator(f(n)) where f(n) = (2*n)!/2^(2*n + 1)(-1)^n, from the Mathematica code.

From Terry D. Grant, May 28 2017: (Start)

|a(n)| = A049606(2n).

a(n) = -numerator(Bernoulli(2n)/z(n)) where Bernoulli(2n) = A000367(n) / A002445(n) and z(n) = A046988(n) / A002432(n) for n > 0. (End) [Corrected by Andrey Zabolotskiy, Sep 23 2021]

Extensions

First term added, offset changed and edited by Johannes W. Meijer, May 15 2009