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A280630
Decimal expansion of Sum_{n>=1} (A001246(n)*A201546(n)) / (A001025(n)*A010050(n)).
1
4, 9, 8, 0, 9, 8, 5, 0, 8, 3, 9, 8, 6, 3, 6, 0, 4, 3, 7, 3, 4, 2, 9, 2, 2, 3, 9, 3, 9, 7, 4, 6, 2, 7, 6, 1, 5, 6, 0, 4, 1, 5, 8, 6, 3, 2, 5, 0, 4, 2, 7, 7, 6, 5, 0, 5, 6, 5, 9, 2, 2, 4, 3, 0, 0, 1, 8, 1, 3, 4, 4, 8, 6, 0, 3, 9, 6, 5, 4, 1
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
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
nonn,cons,base
AUTHOR
John M. Campbell, Jan 06 2017
STATUS
approved