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).
2, 10, 1168, 624240, 114051456, 353810160000, 9271076400000, 86580328116240000, 19402654331894400000, 15000926812307614080000, 437120128035736887168000, 17196604114594832318160000000, 514325437537328572480262784000, 34134351456507030556755674947200000
Offset: 0
Links
- F. Beukers, A note on the irrationality of zeta(2) and zeta(3), Bull. London Math. Soc., Vol. 11, No. 3 (1979), 268-272.
- Wikipedia, Apéry's theorem
Programs
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Maple
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
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Mathematica
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 *)
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