A280630 - OEIS

%I #31 Sep 08 2022 08:46:18

%S 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,

%T 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,

%U 3,4,4,8,6,0,3,9,6,5,4,1

%N Decimal expansion of Sum_{n>=1} (A001246(n)*A201546(n)) / (A001025(n)*A010050(n)).

%C 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.

%H G. C. Greubel, <a href="/A280630/b280630.txt">Table of n, a(n) for n = -1..10000</a>

%H J. M. Campbell, A. Sofo, <a href="https://doi.org/10.1080/10652469.2017.1318874">An integral transform related to series involving alternating harmonic numbers</a>, Integr. Transf. Spec. F., 28 (7) (2017), 547-559.

%H R. B. Paris, <a href="https://zbmath.org/?q=an:1376.33023">Review Zbl 1376.33023</a>, zbMATH 2018.

%F Equals (24 + 16*log(2) - 16*Catalan)/Pi + 8*log(2) - 12, letting Catalan denote Catalan's constant (see A006752).

%F 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.

%e Equals 0.04980985083986360437342922393974627615604158632504...

%t First[RealDigits[(24 + 16 Log[2] - 16 Catalan)/\[Pi] + 8 Log[2] - 12,

%t   10, 80]]

%o (PARI) default(realprecision, 100); (24 + 16*log(2) - 16*Catalan)/Pi + 8*log(2) - 12 \\ _G. C. Greubel_, Aug 25 2018

%o (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

%Y Cf. A001025, A001246, A006752, A010050, A201546.

%K nonn,cons,base

%O -1,1

%A _John M. Campbell_, Jan 06 2017