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A117972
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Numerator of zeta'(-2n), n >= 0.
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9
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1, -1, 3, -45, 315, -14175, 467775, -42567525, 638512875, -97692469875, 9280784638125, -2143861251406875, 147926426347074375, -48076088562799171875, 9086380738369043484375, -3952575621190533915703125
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OFFSET
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0,3
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COMMENTS
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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)
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LINKS
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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
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FORMULA
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a(n) = numerator(f(n)) where f(n) = (2*n)!/2^(2*n + 1)(-1)^n, from the first 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]
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EXAMPLE
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-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), ...
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MAPLE
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# Without rational arithmetic
a := n -> (-1)^n*(2*n)!*2^(add(i, i=convert(n, base, 2))-2*n);
# Peter Luschny, May 02 2009
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MATHEMATICA
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Table[Numerator[(2 n)!/2^(2 n + 1) (-1)^n], {n, 0, 30}]
-(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 *)
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PROG
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(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
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CROSSREFS
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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.
Sequence in context: A359860 A002682 A073595 * A061532 A309453 A360716
Adjacent sequences: A117969 A117970 A117971 * A117973 A117974 A117975
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KEYWORD
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sign,frac
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AUTHOR
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Eric W. Weisstein, Apr 06 2006
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EXTENSIONS
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First term added, offset changed and edited by Johannes W. Meijer, May 15 2009
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STATUS
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approved
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