OFFSET
-1,1
COMMENTS
This Ramanujan-like series may be evaluated in an elegant way in terms of 1/Pi and Catalan's constant, as indicated below in the Formula section.
LINKS
G. C. Greubel, Table of n, a(n) for n = -1..10000
J. M. Campbell, A. Sofo, An integral transform related to series involving alternating harmonic numbers, Integr. Transf. Spec. F., 28 (7) (2017), 547-559.
R. B. Paris, Review Zbl 1376.33023, zbMATH 2018.
FORMULA
Equals (24 + 16*log(2) - 16*Catalan)/Pi + 8*log(2) - 12, letting Catalan denote Catalan's constant (see A006752).
Equals Sum_{n>=0} H'(2n)*C(n)^2/16^n, letting H'(i) denote the i-th alternating harmonic number, and letting C(i) denote the i-th Catalan number.
EXAMPLE
Equals 0.04980985083986360437342922393974627615604158632504...
MATHEMATICA
First[RealDigits[(24 + 16 Log[2] - 16 Catalan)/\[Pi] + 8 Log[2] - 12,
10, 80]]
PROG
(PARI) default(realprecision, 100); (24 + 16*log(2) - 16*Catalan)/Pi + 8*log(2) - 12 \\ G. C. Greubel, Aug 25 2018
(Magma) SetDefaultRealField(RealField(100)); R:=RealField(); (24 + 16*Log(2) - 16*Catalan(R))/Pi(R) + 8*Log(2) - 12; // G. C. Greubel, Aug 25 2018
CROSSREFS
KEYWORD
AUTHOR
John M. Campbell, Jan 06 2017
STATUS
approved