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 A242596 Numerators for partial sums of dilog(1/2). 1

%I

%S 1,9,83,1337,33497,5587,136919,35054939,946522553,946538429,

%T 114531943709,458129108861,77423915447309,38711978428267,

%U 9677996861569,19820539601545337,5728136204565261593,1909378773465525731,689285743475945831291,344642873149232707087

%N Numerators for partial sums of dilog(1/2).

%C The denominators are given as 2*A242597.

%C The limit of r(n) = a(n)/(2*A242597(n)) for n -> infinity is

%C dilog(1/2) = Li_2(1/2) = sum(1/(k^2*2^k),k=1..infinity) = (Pi^2 - 6*(log(2))^2)/12 = 0.582240526465... For the decimal expansion see A076788. See the Abramowitz-Stegun link, p. 1004, 27.7.3 for x=1/2, and the Jolley reference pp. 66-69, (360) (c). See also Jolley, pp. 22-23 (116).

%C This entry was motivated by eight times the sum over the reciprocals of A243456(2*k) for k >= 5. See a comment given there.

%D L. B. W. Jolley, Summation of Series, Dover (1961).

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%F a(n) = numerator(r(n)), with the rational r(n) := sum(1/(k^2*2^k), k=1..n) in lowest terms.

%e The rationals r(n) are, for n=1, ..., 16:

%e 1/2, 9/16, 83/144, 1337/2304, 33497/57600, 5587/9600, 136919/235200, 35054939/60211200, 946522553/1625702400, 946538429/1625702400, 114531943709/196709990400, 458129108861/786839961600, 77423915447309/132975953510400, 38711978428267/66487976755200, 9677996861569/16621994188800, 19820539601545337/34041844098662400.

%Y Cf. A242597, A076788, A243456.

%K nonn,easy

%O 1,2

%A _Wolfdieter Lang_, Jun 16 2014

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Last modified October 20 03:04 EDT 2021. Contains 348099 sequences. (Running on oeis4.)