A303987 -id:A303987 - cp's OEIS frontend

A303988 Triangle read by rows: numerators of c_{n,k}, n >= 0, 0 <= k <= n, used in the proof that Zeta(3) is irrational.

0, 1, 5, 9, 29, 115, 251, 65, 5191, 1039, 2035, 10391, 2077, 72703, 58157, 256103, 259703, 1817471, 1817521, 7270009, 1454021, 28567, 67323, 25243, 389467, 21810107, 47982293, 6854599, 9822481, 9895981, 11132213, 66793523, 11755653433, 2351131157, 30564700141, 30564710941, 78708473, 237497419, 237487619, 23511313481, 23511309071, 61129406407, 5557218637, 61129406447, 244517610353

Offset: 0

Wolfdieter Lang, May 16 2018

The corresponding denominators are given in A303989.
The numerators of the rational triangle c_{n,k} are denoted by T(n,k). The triangle c_{n,k} is used to compute Apéry's sequence of rationals a_n = A059415(n)/A059416(n), satisfying a certain three term recurrence, as a(n) = Sum_{k=0..n} c_{n,k}*(binomial(n+k,k)*binomial(n,k))^2 = Sum_{k=0..n} (T(n,k)/A303989(n,k))*A303987(n,k).
The column k = 0 gives the numerators of Zeta3(n) = A007408(n)/A007409(n), with Zeta3(0) := 0.

			The triangle T(n, k) begins:
  n/k      0       1        2        3           4          5           6
  0:       0
  1:       1       5
  2:       9      29      115
  3:     251      65     5191     1039
  4:    2035   10391     2077    72703       58157
  5:  256103  259703  1817471  1817521     7270009    1454021
  6:   28567   67323    25243   389467    21810107   47982293     6854599
  ...
  row n = 7: 9822481 9895981 11132213 66793523 11755653433 2351131157 30564700141 30564710941,
  row n = 8: 78708473 237497419 237487619 23511313481 23511309071 61129406407 5557218637 61129406447 244517610353,
  row n = 9: 19148110939 19237016539 211601625329 211601801729 2750823224027 42320357851 550164649543 550164651163 37411196140169 37411196579209,
  ...
------------------------------------------------------------------------------
The rational triangle c_{n,k} starts:
  n\k        0            1              2                3               4
  0:        0/1
  1:        1/1          5/4
  2:        9/8         29/24         115/96
  3:      251/216       65/54        5191/4320        1039/864
  4:    2035/1728    10391/8640      2077/1728      72703/60480      58157/48384
  ...
  row n = 5:  256103/216000 259703/216000 1817471/1512000 1817521/1512000 7270009/6048000 1454021/1209600,
  ...

Julian Havil, The Irrationals, Princeton University Press, Princeton and Oxford, 2012, pp. 137-153.

A. van der Poorten, A proof that Euler missed ..., Apery's proof of the irrationality of zeta(3), Math. Intelligencer 1 (1978/79), no. 4, 195-203, c_{n,k} in section 4.
Wikipedia, Apery's theorem

Cf. A005259, A007408/A007409, A059415, A059416, A063007, A303987, A303989.

T(n,k) = numerator(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 27 2025

T(n,k) = numerator(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).

cp's OEIS Frontend

A303988 Triangle read by rows: numerators of c_{n,k}, n >= 0, 0 <= k <= n, used in the proof that Zeta(3) is irrational.

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