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 A214508 Decimal expansion of the series limit sum_{k>=1} (-1)^(k+1) sum_{t=1..k} 1/(t^2*(k+1)^2). 2
 1, 6, 2, 6, 5, 4, 6, 6, 7, 3, 9, 7, 4, 2, 0, 0, 8, 0, 7, 7, 5, 5, 6, 4, 5, 6, 5, 1, 7, 3, 5, 9, 1, 1, 0, 1, 1, 8, 7, 0, 6, 4, 2, 0, 8, 3, 3, 7, 6, 5, 9, 9, 2, 3, 7, 2, 6, 7, 6, 3, 0, 6, 9, 8, 3, 1, 8, 4, 3, 5, 7, 7, 2, 9, 8, 2, 1, 0, 7, 4, 9, 2, 1, 6, 7, 2, 0, 0, 7, 4, 6, 3, 7, 5, 7, 4, 9, 8, 1, 0, 6, 7, 9, 6, 9 (list; constant; graph; refs; listen; history; text; internal format)
 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. FORMULA Equals -4*A099218 +13*Pi^4/288 -7*A002117*log(2)/2+log^2(2)*(Pi^2-log^2(2))/6. 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 Sequence in context: A241033 A216992 A308258 * A343625 A011005 A292178 Adjacent sequences:  A214505 A214506 A214507 * A214509 A214510 A214511 KEYWORD cons,nonn,easy AUTHOR R. J. Mathar, Jul 19 2012 EXTENSIONS More terms from Jean-François Alcover, Feb 12 2013 STATUS approved

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Last modified June 12 10:58 EDT 2021. Contains 344947 sequences. (Running on oeis4.)