cp's OEIS Frontend

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.

%I A262177 #14 Oct 24 2017 09:31:26
%S A262177 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,
%T A262177 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,
%U A262177 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
%N 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.
%C A262177 The similar constant Q_3 = zeta(3) / (Sum_{k>=1} (-1)^(k+1) / (k^3 * binomial(2k, k))) evaluates to 5/2.
%H A262177 G. C. Greubel, <a href="/A262177/b262177.txt">Table of n, a(n) for n = 1..5000</a>
%H A262177 David Bailey, Jonathan Borwein, David Bradley, <a href="http://arxiv.org/abs/math/0505270">Experimental determination of Apéry-like identities for zeta(2n+2)</a>, arXiv:math/0505270 [math.NT], 2005.
%F A262177 Equals 2*zeta(5)/6F5(1,1,1,1,1,1; 3/2,2,2,2,2; -1/4).
%e A262177 2.09486862201003699385024929373294163029675874856778182740127587837438...
%t A262177 Q5 = Zeta[5]/Sum[(-1)^(k+1)/(k^5*Binomial[2k, k]), {k, 1, Infinity}]; RealDigits[Q5, 10, 103] // First
%o A262177 (PARI) zeta(5)/suminf(k=1, (-1)^(k+1)/(k^5*binomial(2*k,k))) \\ _Michel Marcus_, Sep 14 2015
%Y A262177 Cf. A013663.
%Y A262177 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.)
%K A262177 nonn,cons,easy
%O A262177 1,1
%A A262177 _Jean-François Alcover_, Sep 14 2015