This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A303988 #26 May 28 2025 00:53:09
%S A303988 0,1,5,9,29,115,251,65,5191,1039,2035,10391,2077,72703,58157,256103,
%T A303988 259703,1817471,1817521,7270009,1454021,28567,67323,25243,389467,
%U A303988 21810107,47982293,6854599,9822481,9895981,11132213,66793523,11755653433,2351131157,30564700141,30564710941,78708473,237497419,237487619,23511313481,23511309071,61129406407,5557218637,61129406447,244517610353
%N A303988 Triangle read by rows: numerators of c_{n,k}, n >= 0, 0 <= k <= n, used in the proof that Zeta(3) is irrational.
%C A303988 The corresponding denominators are given in A303989.
%C A303988 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).
%C A303988 The column k = 0 gives the numerators of Zeta3(n) = A007408(n)/A007409(n), with Zeta3(0) := 0.
%D A303988 Julian Havil, The Irrationals, Princeton University Press, Princeton and Oxford, 2012, pp. 137-153.
%H A303988 A. van der Poorten, <a href="http://pracownicy.uksw.edu.pl/mwolf/Poorten_MI_195_0.pdf">A proof that Euler missed ..., Apery's proof of the irrationality of zeta(3)</a>, Math. Intelligencer 1 (1978/79), no. 4, 195-203, c_{n,k} in section 4.
%H A303988 Wikipedia, <a href="http://en.wikipedia.org/wiki/Ap%C3%A9ry%27s_theorem">Apery's theorem</a>
%F A303988 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).
%e A303988 The triangle T(n, k) begins:
%e A303988 n/k 0 1 2 3 4 5 6
%e A303988 0: 0
%e A303988 1: 1 5
%e A303988 2: 9 29 115
%e A303988 3: 251 65 5191 1039
%e A303988 4: 2035 10391 2077 72703 58157
%e A303988 5: 256103 259703 1817471 1817521 7270009 1454021
%e A303988 6: 28567 67323 25243 389467 21810107 47982293 6854599
%e A303988 ...
%e A303988 row n = 7: 9822481 9895981 11132213 66793523 11755653433 2351131157 30564700141 30564710941,
%e A303988 row n = 8: 78708473 237497419 237487619 23511313481 23511309071 61129406407 5557218637 61129406447 244517610353,
%e A303988 row n = 9: 19148110939 19237016539 211601625329 211601801729 2750823224027 42320357851 550164649543 550164651163 37411196140169 37411196579209,
%e A303988 ...
%e A303988 ------------------------------------------------------------------------------
%e A303988 The rational triangle c_{n,k} starts:
%e A303988 n\k 0 1 2 3 4
%e A303988 0: 0/1
%e A303988 1: 1/1 5/4
%e A303988 2: 9/8 29/24 115/96
%e A303988 3: 251/216 65/54 5191/4320 1039/864
%e A303988 4: 2035/1728 10391/8640 2077/1728 72703/60480 58157/48384
%e A303988 ...
%e A303988 row n = 5: 256103/216000 259703/216000 1817471/1512000 1817521/1512000 7270009/6048000 1454021/1209600,
%e A303988 ...
%o A303988 (PARI) 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
%Y A303988 Cf. A005259, A007408/A007409, A059415, A059416, A063007, A303987, A303989.
%K A303988 nonn,easy,tabl,frac
%O A303988 0,3
%A A303988 _Wolfdieter Lang_, May 16 2018