A365025 Square array read by antidiagonals: T(n, k) := (k/2)!/k! * ((2*n+1)*k)! * ((2*n+1/2)*k)! / ( (n*k)!^2 * ((n+1/2)*k)!^2 ) for n, k >= 0.

1, 1, 1, 1, 10, 1, 1, 126, 300, 1, 1, 1716, 79380, 11440, 1, 1, 24310, 20612592, 65523780, 485100, 1, 1, 352716, 5318784900, 328206021000, 60634147860, 21841260, 1, 1, 5200300, 1368494343216, 1552041334596844, 5876083665270000, 59774707082376, 1022041020, 1

Offset: 0

Author

Peter Bala, Aug 17 2023

Keywords

• nonn
• tabl
• easy

Comments

Fractional factorials are defined in terms of the gamma function; for example, ((2*n+1/2)*k)! = Gamma(1 + (2*n+1/2)*k).

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.

It is usually assumed that the c's and d's are integers but here we allow for some of the c's and d's to be half-integers. See A276098 for further examples of this type.

Each row sequence of the present table is an integral factorial ratio sequence of height 2.

Conjecture: each row sequence of the table 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.

Examples

			 Square array begins:
 n\k|  0      1               2                    3                      4
  - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  0 |  1      1               1                    1                      1  ...
  1 |  1     10             300                11440                 485100  ...
  2 |  1    126           79380             65523780            60634147860  ...
  3 |  1   1716        20612592         328206021000       5876083665270000  ...
  4 |  1  24310      5318784900     1552041334596844  510031828417402714500  ...
  5 |  1 352716   1368494343216  7108360304262169344 ...
		

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

Crossrefs

Cf. A275652 (row 1), A365026 (row 2), A365027 (row 3).

Cf. A276098, A364506, A364509, A364513, A364518.

Programs

Maple

# display as a square array
T(n, k) := (k/2)!/k! * ((2*n+1)*k)! * ((2*n+1/2)*k)! / ( (n*k)!^2 * ((n+1/2)*k)!^2 ):
seq( print(seq(simplify(T(n, k)), k = 0..10)), n = 0..10);
# display as a sequence
seq( seq(simplify(T(n-k, k)), k = 0..n), n = 0..10);

Python

from itertools import count, islice
from math import factorial
from sympy import factorial2
def A365025_T(n,k): return int(factorial2(k)*factorial(r:=((m:=n<<1)+1)*k)*factorial2(((m<<1)+1)*k)//((factorial(n*k)*factorial2(r))**2*factorial(k)))
def A365025_gen(): # generator of terms
    for n in count(0):
        yield from (A365025_T(n-k,k) for k in range(n+1))
A365025_list = list(islice(A365025_gen(),20)) # Chai Wah Wu, Aug 24 2023

Formula

T(n,k) = Sum_{j = 0..n*k} binomial((2*n+1)*k, n*k-j)^2 * binomial(k+j-1, j).

T(n,k) = binomial((2*n+1)*k,n*k)^2 * hypergeom([k, -n*k, -n*k], [1 + (n+1)*k, 1 + (n+1)*k], 1) = (k/2)!/k! * ((2*n+1)*k)! * ((2*n+1/2)*k)! / ( (n*k)!^2 * ((n+1/2)*k)!^2 ) by Dixon's 3F2 summation theorem.

T(n,k) = [x^(n*k)] ( (1 - x)^(2*n*k) * Legendre_P((2*n+1)*k, (1 + x)/(1 - x)) ).

T(n,k) = k!!*((2*n+1)*k)!*((4*n+1)*k)!!/(k!*((n*k)!*((2*n+1)*k)!!)^2). - Chai Wah Wu, Aug 24 2023