A262177 Decimal expansion of Q_5 = zeta(5) / (Sum_{k>=1} (-1)^(k+1) / (k^5 * binomial(2k, k))), a conjecturally irrational constant defined by an Apéry-like formula.

2, 0, 9, 4, 8, 6, 8, 6, 2, 2, 0, 1, 0, 0, 3, 6, 9, 9, 3, 8, 5, 0, 2, 4, 9, 2, 9, 3, 7, 3, 2, 9, 4, 1, 6, 3, 0, 2, 9, 6, 7, 5, 8, 7, 4, 8, 5, 6, 7, 7, 8, 1, 8, 2, 7, 4, 0, 1, 2, 7, 5, 8, 7, 8, 3, 7, 4, 3, 8, 0, 0, 7, 8, 7, 6, 8, 4, 6, 8, 1, 5, 6, 3, 2, 0, 6, 0, 4, 4, 2, 3, 2, 0, 9, 0, 4, 3, 1, 3, 6, 9, 3, 1

Offset

1,1

Comments

The similar constant Q_3 = zeta(3) / (Sum_{k>=1} (-1)^(k+1) / (k^3 * binomial(2k, k))) evaluates to 5/2.

Links

G. C. Greubel, Table of n, a(n) for n = 1..5000

David Bailey, Jonathan Borwein, David Bradley, Experimental determination of Apéry-like identities for zeta(2n+2), arXiv:math/0505270 [math.NT], 2005.

Formula

Equals 2*zeta(5)/6F5(1,1,1,1,1,1; 3/2,2,2,2,2; -1/4).

Example

2.09486862201003699385024929373294163029675874856778182740127587837438...

Mathematica

Q5 = Zeta[5]/Sum[(-1)^(k+1)/(k^5*Binomial[2k, k]), {k, 1, Infinity}]; RealDigits[Q5, 10, 103] // First

Prog

(PARI) zeta(5)/suminf(k=1, (-1)^(k+1)/(k^5*binomial(2*k, k))) \\ Michel Marcus, Sep 14 2015

Crossrefs

Cf. A013663.

The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692, A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.)

Sequence in context: A168229 A019693 A007493 * A136319 A176057 A272413

Adjacent sequences: A262174 A262175 A262176 * A262178 A262179 A262180

Keyword

nonn,cons,easy

Author

Jean-François Alcover, Sep 14 2015

Status

approved