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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. 34

%I

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

%H G. C. Greubel, <a href="/A262177/b262177.txt">Table of n, a(n) for n = 1..5000</a>

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

%e 2.09486862201003699385024929373294163029675874856778182740127587837438...

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

%o (PARI) zeta(5)/suminf(k=1, (-1)^(k+1)/(k^5*binomial(2*k,k))) \\ _Michel Marcus_, Sep 14 2015

%Y Cf. A013663.

%Y 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 nonn,cons,easy

%O 1,1

%A _Jean-François Alcover_, Sep 14 2015

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