A280630 Decimal expansion of Sum_{n>=1} (A001246(n)*A201546(n)) / (A001025(n)*A010050(n)).

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

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

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

Sequence in context: A145424 A200394 A249272 * A215617 A328229 A366174

Adjacent sequences: A280627 A280628 A280629 * A280631 A280632 A280633

Keyword

nonn,cons,base

Author

John M. Campbell, Jan 06 2017

Status

approved