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 A364519 #19 Aug 14 2023 14:25:24
%S A364519 1,1,0,1,-4,-20,1,0,28,0,1,20,-84,-220,924,1,64,924,0,1820,0,1,140,
%T A364519 12012,48620,16796,-15504,-48620,1,256,60060,2621440,2704156,0,134596,
%U A364519 0,1,420,204204,29745716,608435100,155117520,-3801900,-1184040,2704156,1,640,554268,187432960,15628090140,146028888064,9075135300,0,10518300,0
%N A364519 Square array read by ascending antidiagonals: T(n,k) = [x^(3*k)] ( (1 + x)^(n+3)/(1 - x)^(n-3) )^k for n, k >= 0.
%C A364519 Compare with A364303 and A364518.
%C A364519 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_n(c, d) = (c_1*n)!*(c_2*n)!* ... *(c_K*n)!/ ( (d_1*n)!*(d_2*n)!* ... *(d_L*n)! ) and ask whether it is integral for all n >= 0. The integer L - K is called the height of the sequence. Bober completed the classification of integral factorial ratio sequences of height 1 (see A295431).
%C A364519 Each row of the present table is an integral factorial ratio sequence of height 1. It is usually assumed that the c's and d's are integers but here some of the c's and d's are half-integers. See A276098 and the cross references there for further examples of this type.
%C A364519 It is known that A005810, the unsigned version of row 1, satisfies 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 each row sequence of the table satisfies the same supercongruences.
%H A364519 Peter Bala, <a href="/A276098/a276098.pdf">Some integer ratios of factorials</a>
%H A364519 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 A364519 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hypergeometric_function">Hypergeometric function</a>
%F A364519 T(n,k) = Sum_{j = 0..3*k} binomial((n+3)*k, j)*binomial(n*k-j-1, 3*k-j).
%F A364519 For n >= 3, T(n,k) = binomial(n*k-1,3*k) * hypergeom([-(n+3)*k, -3*k], [1 - n*k], -1) = ((n+3)*k)!*((n-3)*k/2)!/(((n+3)*k/2)!*((n-3)*k)!*(3*k)!) by Kummer's Theorem.
%F A364519 The row generating functions are algebraic functions over the field of rational functions Q(x).
%e A364519 Square array begins:
%e A364519 n\k| 0 1 2 3 4 5
%e A364519 - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
%e A364519 0 | 1 0 -20 0 924 0 ... see A066802
%e A364519 1 | 1 -4 28 -220 1820 -15504 ... see A005810
%e A364519 2 | 1 0 -84 0 16796 0
%e A364519 3 | 1 20 924 48620 2704156 155117520 ... A066802
%e A364519 4 | 1 64 12012 2621440 608435100 146028888064 ... A364520
%e A364519 5 | 1 140 60060 29745716 15628090140 8480843582640 ... A211420
%p A364519 T(n,k) := add( binomial((n+3)*k, j)*binomial(n*k-j-1, 3*k-j), j = 0..3*k):
%p A364519 # display as a square array
%p A364519 seq(print(seq(T(n, k), k = 0..10)), n = 0..10);
%p A364519 # display as a sequence
%p A364519 seq(seq(T(n-k, k), k = 0..n), n = 0..10);
%o A364519 (PARI) T(n,k) = sum(j = 0, 3*k, binomial((n+3)*k, j)*binomial(n*k-j-1, 3*k-j));
%o A364519 lista(nn) = for( n=0, nn, for (k=0, n, print1(T(n-k, k), ", "))); \\ _Michel Marcus_, Aug 13 2023
%Y A364519 Cf. A066802 (row 3, also row 0 unsigned and without 0's), A005810 (row 1 unsigned), A364520 (row 4), A211420 (row 5).
%Y A364519 Cf. A330843, A364303, A364518.
%K A364519 tabl,sign,easy
%O A364519 0,5
%A A364519 _Peter Bala_, Aug 07 2023