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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, 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
0,3
COMMENTS
See A303988 for details, references and links.
FORMULA
T(n, k) = denominator(c_{n,k}), with c_{n,k} = Zeta3(n) + Sum_{m=1..k}((-1)^(m-1))/(2*m*B(n, m)), where Zeta3(n) = Sum_{m=1..n} 1/m^3 = A007408(n)/A007409(n) and B(n, m) = A063007(n, m).
EXAMPLE
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.
CROSSREFS
KEYWORD
nonn,easy,tabl,frac
AUTHOR
Wolfdieter Lang, May 16 2018
STATUS
approved