OFFSET
0,3
COMMENTS
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.
Each row sequence of the present table is an integral factorial ratio sequence of height 2.
It is known that both row 0, 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.
LINKS
J. W. Bober, Factorial ratios, hypergeometric series, and a family of step functions, arXiv:0709.1977 [math.NT], 2007; J. London Math. Soc., 79, Issue 2, (2009), 422-444.
K. Soundararajan, Integral factorial ratios: irreducible examples with height larger than 1, Phil. Trans. Royal Soc., A378: 2018044, 2019.
Wikipedia, Dixon's identity
FORMULA
T(n,k) = Sum_{i = -k..k} (-1)^i * binomial(2*k, k+i) * binomial(2*n*k, n*k+i)^2 (shows that the table entries are integers).
For n >= 1, T(n,k) = (-1)^k * binomial(2*n*k, (n+1)*k)^2 * hypergeom([-2*k, -(n+1)*k, -(n+1)*k], [1 + (n-1)*k, 1 + (n-1)*k], 1) = (2*k)!/k! * ( (2*n*k)! * ((2*n+1)*k)! )/( (n*k)!^2 * ((n+1)*k)!^2 ) by Dixon's 3F2 summation theorem.
T(n,k) = (-1)^k * [x^((n + 1)*k)] ( (1 - x)^(2*(n+1)*k) * Legendre_P(2*n*k, (1 + x)/(1 - x)) ). - Peter Bala, Aug 15 2023
EXAMPLE
Square array begins:
n\k| 0 1 2 3 4 5
- + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
0 | 1 2 6 20 70 252 ...
1 | 1 6 90 1680 34650 756756 ...
2 | 1 40 5880 1101100 229265400 50678855040 ...
3 | 1 350 594594 1299170600 3164045050530 8188909171581600 ...
4 | 1 3528 75088728 2066315135040 63464046079757400 ...
5 | 1 38808 ...
MAPLE
# display as a square array
T(n, k) := (2*k)!/k! * ( (2*n*k)! * ((2*n+1)*k)! )/((n*k)!^2 * ((n+1)*k)!^2):
seq( print(seq(T(n, k), k = 0..10)), n = 0..10);
# display as a sequence
seq( seq(T(n-k, k), k = 0..n), n = 0..10);
CROSSREFS
KEYWORD
AUTHOR
Peter Bala, Jul 27 2023
STATUS
approved