Search: id:A002432
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%I A002432 M4283 N1790
%S A002432 6,90,945,9450,93555,638512875,18243225,325641566250,38979295480125,1531329465290625,
%T A002432 13447856940643125,201919571963756521875,11094481976030578125,564653660170076273671875,
%U A002432 5660878804669082674070015625,62490220571022341207266406250,12130454581433748587292890625
%N A002432 Denominator of zeta(2n)/Pi^(2n).
%C A002432 Also denominators in expansion of Psi(x).
%C A002432 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]
%D A002432 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002432 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002432 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 88.
%D A002432 N. D. Elkies, On the sums Sum_{k = -infinity .. infinity} (4k+1)^(-n), 
               Amer. Math. Monthly, 110 (No. 7, 2003), 561-573.
%D A002432 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.
%D A002432 I. Song, A recursive formula for even order harmonic series, J. Computational 
               and Appl. Math., 21 (1988), 251-256.
%H A002432 T. D. Noe, <a href="b002432.txt">Table of n, a(n) for n=1..100</a>
%H A002432 N. D. Elkies, <a href="http://arXiv.org/abs/math.CA/0101168">On the sums 
               Sum((4k+1)^(-n),k,-inf,+inf)</a>
%H A002432 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               RiemannZetaFunction.html">Link to a section of The World of Mathematics.</
               a>
%F A002432 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
%F A002432 zeta(2n) = Sum_{k >= 1} k^(-2n) = (-1)^(n-1)*B_{2n}*2^(2n-1)*Pi^(2n)/
               (2n)!.
%F A002432 a(n)=-A046988(n)*A010050(n)*A002445(n)/(A009117(n)*A000367(n))
%e A002432 1/6, 1/90, 1/945, 1/9450, 1/93555, 691/638512875, 2/18243225, 3617/325641566250,
               ...
%e A002432 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, ...
%e A002432 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 
               + ...
%p A002432 Zeta(2*n) # then extract denominator of rational part
%Y A002432 Cf. A046988, A006003.
%Y A002432 Sequence in context: A113404 A121607 A100594 this_sequence A091800 A037959 
               A006480
%Y A002432 Adjacent sequences: A002429 A002430 A002431 this_sequence A002433 A002434 
               A002435
%K A002432 nonn,nice,easy,frac
%O A002432 1,1
%A A002432 N. J. A. Sloane (njas(AT)research.att.com).
%E A002432 Formula and link from Henry Bottomley (se16(AT)btinternet.com), Mar 10 
               2000.
%E A002432 Formula corrected by Bjoern Boettcher, May 15, 2003.

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