This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A171485 #26 Aug 02 2025 09:57:30
%S A171485 2,10,1168,624240,114051456,353810160000,9271076400000,
%T A171485 86580328116240000,19402654331894400000,15000926812307614080000,
%U A171485 437120128035736887168000,17196604114594832318160000000,514325437537328572480262784000,34134351456507030556755674947200000
%N A171485 Beukers integral Integral_{y = 0..1} Integral_{x = 0..1} -log(x*y) / (1-x*y) * P_n(2*x-1) * P_n(2*y-1) dx dy = (A(n) + B(n)*zeta(3)) / A003418(n)^3. This sequence gives the values of B(n).
%C A171485 Values of A(n) are given in A171484. P_n(x) are the Legendre Polynomials defined by n!*P_n(x) = (d/dx)^n (x^n*(1-x)^n), see A008316.
%H A171485 F. Beukers, <a href="https://doi.org/10.1112/blms/11.3.268">A note on the irrationality of zeta(2) and zeta(3)</a>, Bull. London Math. Soc., Vol. 11, No. 3 (1979), 268-272.
%H A171485 Wikipedia, <a href="https://en.wikipedia.org/wiki/Ap%C3%A9ry%27s_theorem">Apéry's theorem</a>
%F A171485 a(n) = 2 * A003418(n)^3 * A005259(n). - _Peter Bala_, Aug 01 2025
%p A171485 seq( 2 * lcm(seq(i, i = 1..n))^3 * add(binomial(n,k)^2*binomial(n+k,k)^2, k = 0..n), n = 0..20); # _Peter Bala_, Aug 01 2025
%t A171485 Join[{2}, Table[2*(LCM @@ Range[n])^3 * HypergeometricPFQ[{-n, -n, n + 1, n + 1}, {1, 1, 1}, 1], {n, 1, 20}]] (* _Vaclav Kotesovec_, Aug 02 2025 *)
%Y A171485 Cf. A002117, A003418, A005259, A104684.
%Y A171485 Cf. A008316, A171484.
%K A171485 nonn,easy
%O A171485 0,1
%A A171485 _Max Alekseyev_, Dec 09 2009