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 A303989 #10 May 28 2025 01:11:27
%S A303989 1,1,4,8,24,96,216,54,4320,864,1728,8640,1728,60480,48384,216000,
%T A303989 216000,1512000,1512000,6048000,1209600,24000,56000,21000,324000,
%U A303989 18144000,39916800,5702400,8232000,8232000,9261000,55566000,9779616000,1955923200,25427001600,25427001600,65856000,197568000,197568000,19559232000,19559232000,50854003200,4623091200,50854003200,203416012800
%N A303989 Triangle read by rows: denominators of c_{n,k}, n >= 0, k = 0..n, used in the proof that Zeta(3) is irrational.
%C A303989 See A303988 for details, references and links.
%F A303989 T(n, k) = denominator(c_{n,k}), with c_{n,k} = Zeta3(n) + Sum_{m=1..k} (-1)^(m-1)/(2*m^3*B(n, m)), where Zeta3(n) = Sum_{m=1..n} 1/m^3 = A007408(n)/A007409(n) and B(n, m) = A063007(n, m).
%e A303989 The triangle T(n, k) begins:
%e A303989 n\k 0 1 2 3 4 5 6
%e A303989 0: 1
%e A303989 1: 1 4
%e A303989 2: 8 24 96
%e A303989 3: 216 54 4320 864
%e A303989 4: 1728 8640 1728 60480 48384
%e A303989 5: 216000 216000 1512000 1512000 6048000 1209600
%e A303989 6: 24000 56000 21000 324000 18144000 39916800 5702400
%e A303989 ...
%e A303989 row n = 7: 8232000 8232000 9261000 55566000 9779616000 1955923200 25427001600 25427001600,
%e A303989 row n = 8: 65856000 197568000 197568000 19559232000 19559232000 50854003200 4623091200 50854003200 203416012800,
%e A303989 row n = 9: 16003008000 16003008000 176033088000 176033088000 2288430144000 35206617600 457686028800 457686028800 31122649958400 31122649958400,
%e A303989 ...
%e A303989 For the first rationals c_{n,k} see A303988.
%o A303989 (PARI) T(n,k) = denominator(sum(m=1, n, 1/m^3) + sum(m=1, k, (-1)^(m-1)/(2*m^3*binomial(n,m)*binomial(n+m,m)))) \\ _Jason Yuen_, May 28 2025
%Y A303989 Cf. A007408/A007409, A063007, A303988.
%K A303989 nonn,easy,tabl,frac
%O A303989 0,3
%A A303989 _Wolfdieter Lang_, May 16 2018