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

1, 9, 83, 1337, 33497, 5587, 136919, 35054939, 946522553, 946538429, 114531943709, 458129108861, 77423915447309, 38711978428267, 9677996861569, 19820539601545337, 5728136204565261593, 1909378773465525731, 689285743475945831291, 344642873149232707087

Offset

1,2

Comments

The denominators are given as 2*A242597.

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

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).

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

Links

Table of n, a(n) for n=1..20.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

L. B. W. Jolley, Summation of Series, Dover, 1961

Formula

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

Example

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

Crossrefs

Cf. A242597, A076788, A243456.

Sequence in context: A147960 A155499 A321294 * A180807 A203455 A272582

Adjacent sequences: A242593 A242594 A242595 * A242597 A242598 A242599

Keyword

nonn,easy,changed

Author

Wolfdieter Lang, Jun 16 2014

Status

approved