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A002432
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Denominators of zeta(2n)/Pi^(2n).
(Formerly M4283 N1790)
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5
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6, 90, 945, 9450, 93555, 638512875, 18243225, 325641566250, 38979295480125, 1531329465290625, 13447856940643125, 201919571963756521875, 11094481976030578125, 564653660170076273671875, 5660878804669082674070015625, 62490220571022341207266406250, 12130454581433748587292890625
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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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 (moongerms(AT)wanadoo.fr), Dec 12 2008]
a(0)=2 [From Artur Jasinski (grafix(AT)csl.pl), Mar 11 2010]
a(n) is always divisible by 3 (by the Von Staudt-Clausen theorem, see A002445).
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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).
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LINKS
| N. D. Elkies, On the sums Sum((4k+1)^(-n),k,-inf,+inf)
Eric Weisstein's World of Mathematics, Riemann Zeta Function
J. P. Martin-Flatin and T. D. Noe, Table of n, a(n) for n = 1..250
Index entries for zeta function.
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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 Pérez Herrero, Jan 19 2012.
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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 + ...
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MAPLE
| Zeta(2*n) # then extract denominator of rational part
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MATHEMATICA
| Table[Denominator[Zeta[2 n]/Pi^(2 n)], {n, 1, 30}] (*Artur Jasinski*) [From Artur Jasinski (grafix(AT)csl.pl), Mar 11 2010]
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CROSSREFS
| Cf. A046988, A006003.
Sequence in context: A177283 A121607 A100594 * A091800 A037959 A006480
Adjacent sequences: A002429 A002430 A002431 * A002433 A002434 A002435
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KEYWORD
| nonn,nice,easy,frac
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Formula and link from Henry Bottomley (se16(AT)btinternet.com), Mar 10 2000.
Formula corrected by Bjoern Boettcher, May 15, 2003.
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