OFFSET
0,1
COMMENTS
Also denominators in expansion of Psi(x).
For n >= 1 a(n) is always divisible by 3 (by the von Staudt-Clausen theorem, see A002445).
A comment due to G. Campbell: The original approach taken by Euler was to derive the infinite product for sin(Pi*x)/(Pi*x) equal to (1 - x^2/1^2) (1 - x^2/2^2)(1 - x^2/3^2) ... treating sin(Pi*x)/(Pi*x) as if it were a polynomial. Differentiating the logarithm of both sides and equating coefficients gives all of the zeta function values for 2, 4, 6, 8, .... - M. F. Hasler, Mar 29 2015
Note that 2n+1 divides a(n) for every n. If 2n+1 > 9 is composite, then (2n+1)^2 divides a(n). If 2n+1 is prime, then (2n+1)^2 does not divide a(n). My theorem: for n > 4, (2n+1)^2 divides a(n) if and only if the number 2n+1 is composite. - Thomas Ordowski, Nov 07 2022
REFERENCES
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 88.
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.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
T. D. Noe and J.P. Martin-Flatin, Table of n, a(n) for n = 0..250 (first 100 terms were computed by T. D. Noe).
G. Campbell, Some series like ζ(3), ζ(5), ζ(7). Number Theory group on LinkedIn.com, March 2015.
N. D. Elkies, On the sums Sum((4k+1)^(-n),k,-inf,+inf), arXiv:math/0101168 [math.CA], 2001-2003.
N. D. Elkies, On the sums Sum_{k = -infinity .. infinity} (4k+1)^(-n), Amer. Math. Monthly, 110 (No. 7, 2003), 561-573.
Masato Kobayashi and Shunji Sasaki, Values of zeta-one functions at positive even integers, arXiv:2202.11835 [math.NT], 2022. See p. 4.
J. Sondow and E. W. Weisstein, MathWorld: Riemann Zeta Function
I. Song, A recursive formula for even order harmonic series, J. Computational and Appl. Math., 21 (1988), 251-256.
I. Song, A recursive formula for even order harmonic series, J. Computational and Appl. Math., 21 (1988), 251-256. [Annotated scanned copy]
FORMULA
Sum_{n>=1} 2/(n^2 + 1) = Pi*coth(Pi)-1. 2*Sum_{k>=1} (-1)^(k + 1)/n^(2*k) = 2/(n^2+1). - Shmuel Spiegel (shmualm(AT)hotmail.com), Aug 13 2001
zeta(2n)/(2i * ( log(1-i)-log(1+i) ))^(2n) = zeta(2n)/(-i*log(-1))^(2n). - Eric Desbiaux, Dec 12 2008
zeta(2n) = Sum_{k >= 1} k^(-2n) = (-1)^(n-1)*B_{2n}*2^(2n-1)*Pi^(2n)/(2n)!.
a(n) = sqrt(denominator(Sum_{i>=1} A000005(i)/i^2n)). - Enrique Pérez Herrero, Jan 19 2012
Sum_{k >= 1} zeta(2k)*x^(2k) = (1-Pi*x*cot(Pi*x))/2. - Chris Boyd, Dec 21 2015
a(n) = denominator([x^(2*n)] -x*cot(x)/2). - Peter Luschny, Jun 07 2020
EXAMPLE
(zeta(2n)/Pi^(2n), n = 0, 1, 2, 3, ...) = (-1/2, 1/6, 1/90, 1/945, 1/9450, 1/93555, 691/638512875, 2/18243225, 3617/325641566250, ...), i.e.: zeta(0) = -1/2, zeta(2) = Pi^2/6, zeta(4) = Pi^4/90, zeta(6) = Pi^6/945, zeta(8) = Pi^8/9450, zeta(10) = Pi^10/93555, zeta(12) = Pi^12*691/638512875, ...
In Maple, series(Psi(x),x,20) gives -1*x^(-1) + (-gamma) + 1/6*Pi^2*x + (-Zeta(3))*x^2 + 1/90*Pi^4*x^3 + (-Zeta(5))*x^4 + 1/945*Pi^6*x^5 + (-Zeta(7))*x^6 + 1/9450*Pi^8*x^7 + (-Zeta(9))*x^8 + 1/93555*Pi^10*x^9 + ...
a(5) = 93555 = 10!/(2^9 * B(10)) = 3628800/(512*5/66). - Frank Ellermann, Apr 03 2020
MAPLE
seq(denom(Zeta(2*n)/Pi^(2*n)), n=0..24); # Martin Renner, Sep 07 2016
A002432List := proc(len) series(-x*cot(x)/2, x, 2*len+1):
seq(denom(coeff(%, x, n)), n=0..2*len-1, 2) end:
A002432List(17); # Peter Luschny, Jun 07 2020
MATHEMATICA
Table[Denominator[Zeta[2 n]/Pi^(2 n)], {n, 0, 30}] (* Artur Jasinski, Mar 11 2010 *)
Denominator[Zeta[2*Range[0, 20]]] (* Harvey P. Dale, Sep 05 2013 *)
PROG
(PARI) a(n)=numerator(bestappr(Pi^(2*n)/zeta(2*n))) \\ Requires sufficient realprecision. The standard value of 38 digits yields erroneous values for n>9. \p99 is more than enough to get the 3 lines of displayed data. - M. F. Hasler, Mar 29 2015
(PARI) a002432(n) = denominator(polcoeff((1-x*cotan(x))/2, n*2))
default(seriesprecision, 33); for(i=0, 16, print1(a002432(i), ", ")) \\ Chris Boyd, Dec 21 2015
CROSSREFS
KEYWORD
nonn,nice,easy,frac
AUTHOR
EXTENSIONS
Formula and link from Henry Bottomley, Mar 10 2000
Formula corrected by Bjoern Boettcher, May 15 2003
Corrected and edited by M. F. Hasler, Mar 29 2015
a(0) = 2 prepended by Peter Luschny, Jun 07 2020
STATUS
approved