A046988 Numerators of zeta(2*n)/Pi^(2*n).

-1, 1, 1, 1, 1, 1, 691, 2, 3617, 43867, 174611, 155366, 236364091, 1315862, 6785560294, 6892673020804, 7709321041217, 151628697551, 26315271553053477373, 308420411983322, 261082718496449122051, 3040195287836141605382, 5060594468963822588186

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign
• easy
• frac
• nice

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

Examples

			Numerator(zeta(0)/Pi^0) = -1. - _Artur Jasinski_, Mar 11 2010
		

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.
• I. Song, A recursive formula for even order harmonic series, J. Computational and Appl. Math., 21 (1988), 251-256.
• Wolfram Research, Some values of zeta(n)
• Wolfram Research, A Formula for Zeta(2n)

Crossrefs

Cf. A002432 (denominators), A283301, A266214.

Programs

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 *)