A117972 - OEIS

%I #48 Sep 23 2021 11:45:53

%S 1,-1,3,-45,315,-14175,467775,-42567525,638512875,-97692469875,

%T 9280784638125,-2143861251406875,147926426347074375,

%U -48076088562799171875,9086380738369043484375,-3952575621190533915703125

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

%C 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

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

%C   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

%C From _Andrey Zabolotskiy_, Sep 23 2021: (Start)

%C 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.

%C 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).

%C 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)

%H Vincenzo Librandi, <a href="/A117972/b117972.txt">Table of n, a(n) for n = 0..200</a>

%H Fernando Izaurieta, Ricardo Ramírez and Eduardo Rodríguez, <a href="https://arxiv.org/abs/1106.1648">Dirac Matrices for Chern-Simons Gravity</a>, arXiv:1106.1648 [math-ph], 2011-2012.

%H J. Sondow and E. W. Weisstein, <a href="http://mathworld.wolfram.com/RiemannZetaFunction.html">MathWorld: Riemann Zeta Function</a>

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

%F From _Terry D. Grant_, May 28 2017: (Start)

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

%F 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]

%e -1/4, 3/4, -45/8, 315/4, -14175/8, 467775/8, -42567525/16, ...

%e -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), ...

%p # Without rational arithmetic

%p a := n -> (-1)^n*(2*n)!*2^(add(i,i=convert(n,base,2))-2*n);

%p # _Peter Luschny_, May 02 2009

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

%t -(Numerator[(Table[ BernoulliB[2*n]], {n, 1, 22}] / (Table[[Zeta[2*n]/Pi^(2 n)], {n,1,22}]]) for terms > a(1) (* _Terry D. Grant_, May 28 2017 *)

%o (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

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

%Y From _Johannes W. Meijer_, May 24 2009: (Start)

%Y Cf. A160464, A094665 and A160468.

%Y Absolute values equal row sums of A160468. (End)

%Y Cf. A191657, A191658, A000182, A000041.

%K sign,frac

%O 0,3

%A _Eric W. Weisstein_, Apr 06 2006

%E First term added, offset changed and edited by _Johannes W. Meijer_, May 15 2009