A364506 Square array read by ascending antidiagonals: T(n,k) = (2*k)!/k! * ( (2*n*k)! * ((2*n+1)*k)! )/( (n*k)!^2 * ((n+1)*k)!^2 ).

1, 1, 2, 1, 6, 6, 1, 40, 90, 20, 1, 350, 5880, 1680, 70, 1, 3528, 594594, 1101100, 34650, 252, 1, 38808, 75088728, 1299170600, 229265400, 756756, 924, 1, 453024, 10861066216, 2066315135040, 3164045050530, 50678855040, 17153136, 3432, 1, 5521230, 1721929279200, 3943172216808000

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

Table of n, a(n) for n=0..39.

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

A000984 (row 0), A006480 (row 1), A364507 (row 2), A364508 (row 3). Cf. A364303, A364509, A365025.

Sequence in context: A090582 A079641 A373660 * A222864 A232433 A271881

Adjacent sequences: A364503 A364504 A364505 * A364507 A364508 A364509

Keyword

nonn,tabl,easy

Author

Peter Bala, Jul 27 2023

Status

approved