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

0,7

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.

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)

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

Cf. A002432 (denominators), A283301, A266214.

Sequence in context: A141590 A276594 A283301 * A189683 A029825 A180320

Adjacent sequences: A046985 A046986 A046987 * A046989 A046990 A046991

Keyword

sign,easy,frac,nice

Author

N. J. A. Sloane

Status

approved