A214508 Decimal expansion of the series limit Sum_{k>=1} (-1)^(k+1) Sum_{t=1..k} 1/(t^2*(k+1)^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

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...

Sofo and Choi (2022) called it the Rutledge-Douglass-Raynor constant after the American mathematicians George Rutledge (1881-1940), Raymond Donald Douglass (1894-1978), and George Emil Raynor (1895-1975). The constant was introduced by Rutledge and Douglass in 1934 and was denoted by A_4. - Amiram Eldar, Oct 13 2025

Links

Table of n, a(n) for n=0..104.

David H. Bailey, Jonathan M. Borwein, and Roland Girgensohn, Experimental evaluation of Euler sums, Exp. Math. 3 (1994), 17-30; alternative link; variable alpha(2,2).

Mark W. Coffey and Nicholas Lubbers, On generalized harmonic number sums, Appl. Math. Comput., Vol. 217, No. 2 (2010), pp. 689-698.

G. E. Raynor, On Serret's integral formula, Bull. Am. Math. Soc., Vol. 45 (1939), pp. 911-917.

George Rutledge and R. D. Douglass, Evaluation of Integral_{0..1} (log u / u) log^2(1 + u) du and Related Definite Integrals, The American Mathematical Monthly, Vol. 41, No. 1 (1934), pp. 29-36.

George Rutledge and R. D. Douglass, Table of definite integrals, Am. Math. Monthly, Vol. 45, No. 8 (1938), 525-530, variable A_4.

R. Sitaramachandrarao, A formula of S. Ramanujan, Journal of Number Theory, Vol. 25, No. 1 (1987), pp. 1-19.

Anthony Sofo and Junesang Choi, Extension of the four Euler sums being linear with parameters and series involving the zeta functions, Journal of Mathematical Analysis and Applications, Vol. 515, No. 1 (2022), Article 126370.

Formula

Equals -4*A099218 + 13*Pi^4/288 - 7*A002117*log(2)/2+log^2(2)*(Pi^2-log^2(2))/6.

From Amiram Eldar, Oct 13 2025: (Start)

Two formulas from Rutledge and Douglass (1934):

Equals Pi^4/288 + Integral_{x=0..1} (log(x)/x) * log(1+x)^2 dx.

Equals -31*Pi^4/480 + Integral_{x=0..Pi} x * log(2*cos(x/2))^2 dx.

Equals 3*zeta(4)/4 - 4*G(1), where G(1) = Sum_{r>=1} ((1/(2*r)^3) * Sum_{k=1..r} 1/(2*k-1)) = (Pi/4) * Sum_{r>=0} (-1)^r/(4*r+1)^3 - (Pi/(3*sqrt(3))) * Sum_{r>=0} 1/(4*r+1)^3 (Sitaramachandrarao, 1987). (End)

Example

0.16265466739742008077556456517359110118706420833765...

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

Cf. A002117, A007406, A099218.

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