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A257045 Denominators of Apéry's rational approximations p_n/q_n to zeta(3).

%I A257045 #16 Feb 16 2025 08:33:25
%S A257045 1,5,292,52020,9504288,29484180000,17168660000,801669704780000,
%T A257045 35930841355360000,1250077234358967840000,36426677336311407264000,
%U A257045 11464402743063221545440000,42860453128110714373355232000
%N A257045 Denominators of Apéry's rational approximations p_n/q_n to zeta(3).
%H A257045 Seiichi Manyama, <a href="/A257045/b257045.txt">Table of n, a(n) for n = 0..358</a>
%H A257045 Stéphane Fischler, <a href="https://eudml.org/doc/252133">Irrationalité de valeurs de zêta.</a> Séminaire Bourbaki (2002-2003) Vol. 45, pp. 27-62 [in French]
%H A257045 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/AperysConstant.html">Apéry's Constant</a>
%H A257045 Wikipedia, <a href="http://en.wikipedia.org/wiki/Ap%C3%A9ry%27s_constant">Apéry's constant</a>
%F A257045 See Mathematica script.
%e A257045 0, 6/5,  351/292,  62531/52020, 11424695/9504288, 35441662103/29484180000, ...
%e A257045 0, 1.2, 1.202..., 1.2020569...,   1.202056903...,        1.20205690316..., ...
%t A257045 p[n_] := Sum[Binomial[n+k, k]^2*Binomial[n, k]^2*(Sum[1/m^3, {m, 1, n}] + Sum[ (-1)^(m-1)/(2*m^3*Binomial[n, m]*Binomial[m+n, m]), {m, 1, k}]), {k, 0, n}]; q[n_] := Sum[Binomial[n+k, k]^2*Binomial[n, k]^2, {k, 0, n}]; Table[p[n]/q[n], {n, 0, 12}] // Denominator
%Y A257045 Cf. A002117, A005259, A059415, A059416.
%K A257045 nonn,frac
%O A257045 0,2
%A A257045 _Jean-François Alcover_, Apr 15 2015