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A002432 Denominators of zeta(2n)/Pi^(2n).
(Formerly M4283 N1790)
5
6, 90, 945, 9450, 93555, 638512875, 18243225, 325641566250, 38979295480125, 1531329465290625, 13447856940643125, 201919571963756521875, 11094481976030578125, 564653660170076273671875, 5660878804669082674070015625, 62490220571022341207266406250 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Also denominators in expansion of Psi(x).

zeta(2n)/(2i * ( ln(1-i)-ln(1+i) ))^(2n) = zeta(2n)/(-i*ln(-1))^(2n) [From Eric Desbiaux, Dec 12 2008]

a(0)=2 [From Artur Jasinski, Mar 11 2010]

a(n) is always divisible by 3 (by the Von Staudt-Clausen theorem, see A002445).

REFERENCES

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 88.

N. D. Elkies, On the sums Sum_{k = -infinity .. infinity} (4k+1)^(-n), Amer. Math. Monthly, 110 (No. 7, 2003), 561-573.

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.

I. Song, A recursive formula for even order harmonic series, J. Computational and Appl. Math., 21 (1988), 251-256.

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 = 1..250 (first 100 terms were computed by T. D. Noe)

N. D. Elkies, On the sums Sum((4k+1)^(-n),k,-inf,+inf)

J. Sondow and E. W. Weisstein, MathWorld: Riemann Zeta Function

Index entries for zeta function.

FORMULA

Sum[2/(n^2 + 1), {n, 1, Infinity}] = Pi*Coth[Pi]-1. 2*Sum[(-1)^(k + 1)/n^(2*k), {k, 1, Infinity}] = 2/(n^2+1). - Shmuel Spiegel (shmualm(AT)hotmail.com), Aug 13 2001

zeta(2n) = Sum_{k >= 1} k^(-2n) = (-1)^(n-1)*B_{2n}*2^(2n-1)*Pi^(2n)/(2n)!.

a(n)=-A046988(n)*A010050(n)*A002445(n)/(A009117(n)*A000367(n))

a(n)=sqrt(denominator(sum_{i=1...infinity} A000005(i)/i^2n)) - Enrique PĂ©rez Herrero, Jan 19 2012.

EXAMPLE

1/6, 1/90, 1/945, 1/9450, 1/93555, 691/638512875, 2/18243225, 3617/325641566250,...

zeta(2) = Pi^2/6, zeta(4) = Pi^4/90, zeta(6) = Pi^6/945, Pi^8/9450, P{i^10/93555, 691*Pi^12/638512875, ...

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 + ...

MAPLE

Zeta(2*n) # then extract denominator of rational part

MATHEMATICA

Table[Denominator[Zeta[2 n]/Pi^(2 n)], {n, 1, 30}] (* Artur Jasinski, Mar 11 2010 *)

Denominator[Zeta[2*Range[20]]] (* Harvey P. Dale, Sep 05 2013 *)

CROSSREFS

Cf. A046988, A006003.

Sequence in context: A177283 A121607 A100594 * A091800 A037959 A201073

Adjacent sequences:  A002429 A002430 A002431 * A002433 A002434 A002435

KEYWORD

nonn,nice,easy,frac

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Formula and link from Henry Bottomley, Mar 10 2000.

Formula corrected by Bjoern Boettcher, May 15, 2003.

STATUS

approved

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Last modified September 30 17:45 EDT 2014. Contains 247475 sequences.