1,1

Also denominators in expansion of Psi(x).

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.

T. D. Noe, Table of n, a(n) for n=1..100

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

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

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))

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, ...

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 + ...

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

Cf. A046988, A006003.

Sequence in context: A113404 A121607 A100594 this_sequence A091800 A037959 A006480

Adjacent sequences: A002429 A002430 A002431 this_sequence A002433 A002434 A002435

nonn,nice,easy,frac

njas

Formula and link from Henry Bottomley (se16(AT)btinternet.com), Mar 10 2000.

Formula corrected by Bjoern Boettcher, May 15, 2003.