A303989 Triangle read by rows: denominators of c_{n,k}, n >= 0, k = 0..n, used in the proof that Zeta(3) is irrational.
1, 1, 4, 8, 24, 96, 216, 54, 4320, 864, 1728, 8640, 1728, 60480, 48384, 216000, 216000, 1512000, 1512000, 6048000, 1209600, 24000, 56000, 21000, 324000, 18144000, 39916800, 5702400, 8232000, 8232000, 9261000, 55566000, 9779616000, 1955923200, 25427001600, 25427001600, 65856000, 197568000, 197568000, 19559232000, 19559232000, 50854003200, 4623091200, 50854003200, 203416012800
Offset: 0
Examples
The triangle T(n, k) begins:
n\k 0 1 2 3 4 5 6
0: 1
1: 1 4
2: 8 24 96
3: 216 54 4320 864
4: 1728 8640 1728 60480 48384
5: 216000 216000 1512000 1512000 6048000 1209600
6: 24000 56000 21000 324000 18144000 39916800 5702400
...
row n = 7: 8232000 8232000 9261000 55566000 9779616000 1955923200 25427001600 25427001600,
row n = 8: 65856000 197568000 197568000 19559232000 19559232000 50854003200 4623091200 50854003200 203416012800,
row n = 9: 16003008000 16003008000 176033088000 176033088000 2288430144000 35206617600 457686028800 457686028800 31122649958400 31122649958400,
...
For the first rationals c_{n,k} see A303988.
Programs
-
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
Comments