OFFSET
0,7
COMMENTS
Equivalently, numerator of (-1)^(n+1)*2^(2*n-1)*Bernoulli(2*n)/(2*n)!. - Lekraj Beedassy, Jun 26 2003
An old name erroneously included "Numerators of Taylor series expansion of log(x/sin(x))"; that is now submitted as a distinct sequence A283301. - Vladimir Reshetnikov, Mar 04 2017
REFERENCES
L. V. Ahlfors, Complex Analysis, McGraw-Hill, 1979, p. 205
T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 222, series for log(H(x)/x).
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 88.
CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 42.
A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 84.
LINKS
J.P. Martin-Flatin, Table of n, a(n) for n = 0..250
Masato Kobayashi and Shunji Sasaki, Values of zeta-one functions at positive even integers, arXiv:2202.11835 [math.NT], 2022. See p. 4.
Ellise Parnoff and A. Raghuram, Ramanujan's congruence primes, arXiv:2403.03345 [math.NT], 2024.
Andres Robles-Navarro, Shaun Cooper, Andreas A. Buchheit, Jonathan Busse, Antony Burrows, Odile Smits, and Peter Schwerdtfeger, Exact lattice summations for Lennard-Jones potentials coupled to a three-body Axilrod-Teller-Muto term applied to cuboidal phase transitions, arXiv:2504.07338 [cond-mat.mtrl-sci], 2025. See p. 36 (Appendix A).
Iickho Song, A recursive formula for even order harmonic series, J. Comput. Appl. Math., 21 (1988), 251-256.
Wolfram Research, Some values of zeta(n).
Wolfram Research, A Formula for Zeta(2n).
EXAMPLE
Numerator(zeta(0)/Pi^0) = -1. - Artur Jasinski, Mar 11 2010
MAPLE
seq(numer(Zeta(2*n)/Pi^(2*n)), n=1..24); # Martin Renner, Sep 07 2016
MATHEMATICA
Table[Numerator[Zeta[2 n]/Pi^(2 n)], {n, 0, 30}] (* Artur Jasinski, Mar 11 2010 *)
CROSSREFS
KEYWORD
sign,easy,frac,nice
AUTHOR
STATUS
approved