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

Author

Wolfdieter Lang, May 16 2018

Keywords

• nonn
• easy
• tabl
• frac

Comments

See A303988 for details, references and links.

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.
		

Crossrefs

Cf. A007408/A007409, A063007, 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

Formula

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).