A303988 - OEIS

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

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OFFSET

0,3

COMMENTS

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.

REFERENCES

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

LINKS

Table of n, a(n) for n=0..44.

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

FORMULA

T(n,k) = numerator(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: 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,

...

CROSSREFS

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

Sequence in context: A006131 A352008 A270595 * A242329 A343420 A214902

Adjacent sequences: A303985 A303986 A303987 * A303989 A303990 A303991

KEYWORD

nonn,easy,tabl,frac

AUTHOR

Wolfdieter Lang, May 16 2018

STATUS

approved