OFFSET
0,3
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).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Fernando Izaurieta, Ricardo RamÃrez and Eduardo RodrÃguez, Dirac Matrices for Chern-Simons Gravity, arXiv:1106.1648 [math-ph], 2011-2012.
J. Sondow and E. W. Weisstein, MathWorld: Riemann Zeta Function
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]
EXAMPLE
-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), ...
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}]
PROG
(Maxima) L:taylor(1/x*sin(sqrt(x))^2, x, 0, 15); makelist(denom(coeff(L, x, n)), n, 0, 15); // Vladimir Kruchinin, May 30 2011
CROSSREFS
From Johannes W. Meijer, May 24 2009: (Start)
Absolute values equal row sums of A160468. (End)
KEYWORD
sign,frac
AUTHOR
Eric W. Weisstein, Apr 06 2006
EXTENSIONS
First term added, offset changed and edited by Johannes W. Meijer, May 15 2009
STATUS
approved