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%I A364518 #23 Jul 09 2024 02:21:46
%S A364518 1,1,-2,1,0,6,1,6,-10,-20,1,16,70,0,70,1,30,630,924,198,-252,1,48,
%T A364518 2310,28672,12870,0,924,1,70,6006,204204,1385670,184756,-4420,-3432,1,
%U A364518 96,12870,860160,19122246,69206016,2704156,0,12870,1,126,24310,2704156,130378950,1848483780,3528923580,40116600,104006,-48620
%N A364518 Square array read by ascending antidiagonals: T(n,k) = [x^(2*k)] ( (1 + x)^(n+2)/(1 - x)^(n-2) )^k for n, k >= 0.
%C A364518 Compare with A364303 and A364519.
%C A364518 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). Soundararajan gives many examples of two-parameter families of integral factorial ratio sequences of height 2.
%C A364518 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 A364518 It is known that the unsigned version of row 0 (the central binomial numbers A000984) and row 2 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.
%H A364518 Peter Bala, <a href="/A276098/a276098.pdf">Some integer ratios of factorials</a>
%H A364518 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 A364518 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 A364518 Wikipedia, <a href="https://en.wikipedia.org/wiki/Dixon%27s_identity">Dixon's identity</a>
%H A364518 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hypergeometric_function">Hypergeometric function</a>
%F A364518 T(n,k) = Sum_{j = 0..2*k} binomial((n+2)*k, j)*binomial(n*k-j-1, 2*k-j).
%F A364518 T(2,k) = binomial(4*k,2*k).
%F A364518 For n >= 3, T(n,k) = binomial(n*k-1,2*k) * hypergeom([-(n+2)*k, -2*k], [1 - n*k], -1) except when (n,k) = (3,1).
%F A364518 For n >= 2, T(n,k) = ((n+2)*k)!*((n-2)*k/2)!/(((n+2)*k/2)!*((n-2)*k)!*(2*k)!) by Kummer's Theorem.
%F A364518 T(n,k) = [x^k] (1 - x)^(2*k) * Chebyshev_T(n*k, (1 + x)/(1 - x)).
%F A364518 T(n,k) = Sum_{j = 0..k} binomial(2*n*k, 2*j)*binomial((n-1)*k-j-1, k-j).
%F A364518 For n >= 3, T(n,k) = binomial((n-1)*k-1,k) * hypergeom([-n*k, -k, -n*k + 1/2], [1 - (n-1)*k, 1/2], 1).
%F A364518 The row generating functions are algebraic functions over the field of rational functions Q(x).
%e A364518  Square array begins:
%e A364518  n\k|  0   1      2        3           4             5
%e A364518   - + - - - - - - - - - - - - - - - - - - - - - - - - -
%e A364518   0 |  1  -2      6      -20          70          -252   ... see A000984
%e A364518   1 |  1   0    -10        0         198             0   ... see A211419
%e A364518   2 |  1   6     70      924       12870        184756   ... A001448
%e A364518   3 |  1  16    630    28672     1385670      69206016   ... A091496
%e A364518   4 |  1  30   2310   204204    19122246    1848483780   ... A061162
%e A364518   5 |  1  48   6006   860160   130378950   20392706048   ... A276098
%e A364518   6 |  1  70  12870  2704156   601080390  137846528820   ... A001448 bisected
%e A364518   7 |  1  96  24310  7028736  2149374150  678057476096   ... A276099
%p A364518 T(n,k) = add( binomial((n+2)*k, j)*binomial(n*k-j-1, 2*k-j), j = 0..2*k):
%p A364518 # display as a square array
%p A364518 seq(print(seq(T(n, k), k = 0..10)), n = 0..10);
%p A364518 # display as a sequence
%p A364518 seq(seq(T(n-k, k), k = 0..n), n = 0..10);
%o A364518 (PARI) T(n,k) = sum(j = 0, 2*k, binomial((n+2)*k, j)*binomial(n*k-j-1, 2*k-j));
%o A364518 lista(nn) = for( n=0, nn, for (k=0, n, print1(T(n-k, k), ", "))); \\ _Michel Marcus_, Aug 13 2023
%Y A364518 Cf. A000984 (row 0 unsigned), A211419 (row 1 unsigned without 0's), A001448 (row 2), A091496 (row 3), A061162 (row 4), A276098 (row 5), A001448 bisected (row 6), A276099 (row 7).
%Y A364518 Cf. A330843, A364303, A364506, A364509, A364513, A364519.
%K A364518 tabl,sign,easy
%O A364518 0,3
%A A364518 _Peter Bala_, Aug 07 2023

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