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 A364303 #17 Jul 24 2023 15:09:46
%S A364303 1,1,-2,1,0,6,1,4,-6,-20,1,10,36,0,70,1,18,300,400,90,-252,1,28,1050,
%T A364303 11440,4900,0,924,1,40,2646,77616,485100,63504,-1680,-3432,1,54,5544,
%U A364303 316540,6370650,21841260,853776,0,12870,1,70,10296,972400,42031990,554822268,1022041020,11778624,34650,-48620
%N A364303 Square array read by ascending antidiagonals: T(n,k) = [x^k] (1 - x)^(2*k) * Legendre_P(n*k, (1 + x)/(1 - x)) for n, k >= 0.
%C A364303 The first row of the table is a signed version of the central binomial coefficients A000984. The central binomial coefficients 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 (see Meštrović, equation 39). We conjecture that each row sequence of the table satisfies the same supercongruences.
%H A364303 R. 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 A364303 Wikipedia, <a href="https://en.wikipedia.org/wiki/Dixon%27s_identity">Dixon's identity</a>
%F A364303 T(n,k) = Sum_{i = 0..k} binomial(n*k, k-i)^2 * binomial((n-2)*k+i-1, i).
%F A364303 For n >= 2, T(n,k) = binomial((n-1)*k, k)^2 * hypergeom([a, b, b], [1 + a - b, 1 + a - b], 1), where a = (n - 3)*k and b = -k.
%F A364303 For n >= 3, T(n,k) = ((n - 1)*k)! * ((n + 1)*k/2)! * ((n - 3)*k/2)! / ( ((n - 1)*k/2)!^2 * k!^2 * ((n - 3)*k)! ) by Dixon's 3F2 summation theorem, where fractional factorials are defined in terms of the gamma function.
%e A364303 Square array begins:
%e A364303 n\k| 0 1 2 3 4 5
%e A364303 - + - - - - - - - - - - - - - - - - - - - - - - - - -
%e A364303 0 | 1 -2 6 -20 70 -252 ... (-1)^k*A000984(k)
%e A364303 1 | 1 0 -6 0 90 0 ... A245086
%e A364303 2 | 1 4 36 400 4900 63504 ... A002894
%e A364303 3 | 1 10 300 11440 485100 21841260 ... A275652
%e A364303 4 | 1 18 1050 77616 6370650 554822268 ... A275653
%e A364303 5 | 1 28 2646 316540 42031990 5921058528 ... A275654
%e A364303 6 | 1 40 5544 972400 189290920 39089615040 ... A275655
%e A364303 7 | 1 54 10296 2484000 665091000 188907932304 ... A364304
%e A364303 8 | 1 70 17550 5567380 1960044750 732012601320 ... A364305
%p A364303 T(n,k) := coeff(series( (1 - x)^(2*k) * LegendreP(n*k, (1 + x)/(1 - x)), x, 11), x, k):
%p A364303 # display as a square array
%p A364303 seq(print(seq(T(n, k), k = 0..10)), n = 0..10);
%p A364303 # display as a sequence
%p A364303 seq(seq(T(n-k, k), k = 0..n), n = 0..10);
%Y A364303 Cf. A000984 (row 0 unsigned), A245086 (row 1), A002894 (row 2), A275652 (row 3), A275653 (row 4), A275654 (row 5), A275655 (row 6), A364304 (row 7), A364305 (row 8).
%Y A364303 Cf. A364113, A364298
%K A364303 sign,tabl,easy
%O A364303 0,3
%A A364303 _Peter Bala_, Jul 19 2023