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 A364509 #30 Oct 13 2024 07:07:39
%S A364509 1,1,4,1,6,36,1,16,90,400,1,50,784,1680,4900,1,168,8910,48400,34650,
%T A364509 63504,1,588,113256,2011100,3312400,756756,853776,1,2112,1528436,
%U A364509 96993024,503909070,240374016,17153136,11778624,1,7722,21395520,5056527000,92279796840,133954543800,18116083216
%N A364509 Square array read by ascending antidiagonals: T(n,k) = (2*k)!/k!^2 * ( (2*n*k)! * ((n + 2)*k)! )/( (n*k)! * ((n + 1)*k)!^2 ) for n, k > = 0.
%C A364509 Given two sequences of integers c = (c_1, c_2, ..., c_K) and d = (d_1, d_2, ..., d_L) where c_1 + ... + c_K = d_1 + ... + d_L we can define the factorial ratio sequence u_k(c, d) = (c_1*k)!*(c_2*k)!* ... *(c_K*k)!/ ( (d_1*k)!*(d_2*k)!* ... *(d_L*k)! ) and ask whether it is integral for all k >= 0. The integer L - K is called the height of the sequence. Bober completed the classification of integral factorial ratio sequences of height 1. Soundararajan gives many examples of two-parameter families of integral factorial ratio sequences of height 2.
%C A364509 Each row sequence of the present table is an integral factorial ratio sequence of height 2.
%C A364509 It is known that both row 0, the squares of the central binomial numbers, and row 1, the de Bruijn numbers, satisfy the supercongruences u(n*p^r) == u(n*p^(r-1)) (mod p^(3*r)) for all primes p >= 5 and all positive integers n and r. We conjecture that all the row sequences of the table satisfy the same supercongruences [added Oct 11 2024: follows from Meštrović, Section 6, equation 39, since T(n, k) = binomial(2*k, k) * binomial(2*n*k, n*k) * binomial((n+2)*k, k)/binomial((n+1)*k, k)].
%H A364509 Winston de Greef, <a href="/A364509/b364509.txt">Table of n, a(n) for n = 0..3240</a> (80 antidiagonals)
%H A364509 J. W. Bober, <a href="https://arxiv.org/abs/0709.1977">Factorial ratios, hypergeometric series, and a family of step functions</a>, arXiv:0709.1977 [math.NT], 2007; J. London Math. Soc. (2) 79 2009, 422-444.
%H A364509 Romeo Meštrović, <a href="https://arxiv.org/abs/1111.3057">Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2012)</a>, arXiv:1111.3057 [math.NT], (2011).
%H A364509 K. Soundararajan, <a href="http://doi.org/10.1098/rsta.2018.0444">Integral factorial ratios: irreducible examples with height larger than 1</a>, Phil. Trans. Royal Soc., A378: 2018044, 2019.
%H A364509 Wikipedia, <a href="https://en.wikipedia.org/wiki/Dixon%27s_identity">Dixon's identity</a>
%F A364509 T(n,k) = Sum_{i = -k..k} (-1)^i * binomial(2*k, k+i)^2 * binomial(2*n*k, n*k+i) (shows that the table entries are integers).
%F A364509 For n >= 1, T(n,k) = (-1)^k * binomial(2*n*k, (n+1)*k)^2 * hypergeom([-2*k, -2*k, -(n+1)*k], [1, 1 + (n-1)*k], 1) = (2*k)!/k!^2 * ( (2*n*k)! * ((n + 2)*k)! )/( (n*k)! * ((n + 1)*k)!^2 ) by Dixon's 3F2 summation theorem.
%F A364509 T(n,k) = (-1)^(n*k) * [x^((n+1)*k)] ( (1 - x)^(2*(n+1)*k) * Legendre_P(2*k, (1 + x)/(1 - x)) ). - _Peter Bala_, Aug 14 2023
%e A364509 Square array begins:
%e A364509 n\k| 0 1 2 3 4 5
%e A364509 - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
%e A364509 0 | 1 4 36 400 4900 63504 ... A002894
%e A364509 1 | 1 6 90 1680 34650 756756 ... A006480
%e A364509 2 | 1 16 784 48400 3312400 240374016 ... A364510
%e A364509 3 | 1 50 8910 2011100 503909070 133954543800 ... A364511
%e A364509 4 | 1 168 113256 96993024 92279796840 93172920645168 ...
%e A364509 5 | 1 588 1528436 5056527000 18592935952500 72567511917065088 ...
%p A364509 # display as a square array
%p A364509 T(n,k) := (2*k)!/k!^2 * ( (2*n*k)! * ((n + 2)*k)! )/( (n*k)! * ((n + 1)*k)!^2 ):
%p A364509 seq( print(seq(T(n,k), k = 0..10)), n = 0..10):
%p A364509 # display as a sequence
%p A364509 seq( seq(T(n-k,k), k = 0..n), n = 0..10);
%o A364509 (PARI) T(n,k) = (2*k)!/k!^2 * ( (2*n*k)! * ((n + 2)*k)! )/( (n*k)! * ((n + 1)*k)!^2 ) \\ _Winston de Greef_, Oct 05 2023
%Y A364509 A002894 (row 0), A006480 (row 1), A364510 (row 3), A364511 (row 4).
%Y A364509 Cf. A364303, A364506.
%K A364509 nonn,tabl,easy
%O A364509 0,3
%A A364509 _Peter Bala_, Jul 28 2023