OFFSET
0,3
COMMENTS
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.
LINKS
R. Meštrović, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862--2012), arXiv:1111.3057 [math.NT], 2011.
Wikipedia, Dixon's identity
FORMULA
T(n,k) = Sum_{i = 0..k} binomial(n*k, k-i)^2 * binomial((n-2)*k+i-1, i).
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.
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.
EXAMPLE
Square array begins:
n\k| 0 1 2 3 4 5
- + - - - - - - - - - - - - - - - - - - - - - - - - -
0 | 1 -2 6 -20 70 -252 ... (-1)^k*A000984(k)
1 | 1 0 -6 0 90 0 ... A245086
2 | 1 4 36 400 4900 63504 ... A002894
3 | 1 10 300 11440 485100 21841260 ... A275652
4 | 1 18 1050 77616 6370650 554822268 ... A275653
5 | 1 28 2646 316540 42031990 5921058528 ... A275654
6 | 1 40 5544 972400 189290920 39089615040 ... A275655
7 | 1 54 10296 2484000 665091000 188907932304 ... A364304
8 | 1 70 17550 5567380 1960044750 732012601320 ... A364305
MAPLE
T(n, k) := coeff(series( (1 - x)^(2*k) * LegendreP(n*k, (1 + x)/(1 - x)), x, 11), x, k):
# display as a square array
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 19 2023
STATUS
approved