OFFSET
0,2
COMMENTS
Equals the alternating sum over (-1)^(k+1)*H_k^(2)/(k+1)^2, where H_k^(2) is the harmonic sum over inverse squares, H_k^(2) = sum_{t=1..k} 1/t^2 = 1, 5/4, 49/36, 205/144, 5269/3600,..., see A007406. The sum over H_k^(2)/(k+1)^2, over the absolute values, is Pi^4/120 = 0.811742425283353...
LINKS
D. H. Bailey, J. M. Borwein, R. Girgensohn, Experimental evaluation of Euler sums, Exp. Math. 3 (1994) 17, variable alpha(2,2)
G. Rutledge, R. D. Douglass, Table of definite integrals, Am. Math. Monthly 45 (1938) 525, variable A_4.
EXAMPLE
0.162654667397420080...
MAPLE
a099218 := polylog(4, 1/2) ;
-4*a099218+13*Pi^4/288-7/2*Zeta(3)*log(2)+Pi^2/6*(log(2))^2-(log(2))^4/6 ;
evalf(%) ;
MATHEMATICA
NSum[(-1)^(k + 1)*HarmonicNumber[k, 2]/(k + 1)^2, {k, 1, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> 110] // RealDigits[#, 10, 105] & // First (* or, from formula: *) 13*Pi^4/288 + 1/6*Pi^2*Log[2]^2 - 1/6*Log[2]*(Log[2]^3 + 21*Zeta[3]) - 4*PolyLog[4, 1/2] // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Mar 06 2013 *)
PROG
(PARI) 13*Pi^4/288 + 1/6*Pi^2*log(2)^2 - 1/6*log(2)*(log(2)^3 + 21*zeta(3)) - 4*polylog(4, 1/2) \\ Charles R Greathouse IV, Jul 18 2014
CROSSREFS
KEYWORD
AUTHOR
R. J. Mathar, Jul 19 2012
EXTENSIONS
More terms from Jean-François Alcover, Feb 12 2013
STATUS
approved