|
| |
|
|
A303988
|
|
Triangle read by rows: numerators of c_{n,k}, n >= 0, 0 <= k <= n, used in the proof that Zeta(3) is irrational.
|
|
1
|
|
|
|
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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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
|
|
|
|
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
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
