A364519 Square array read by ascending antidiagonals: T(n,k) = [x^(3*k)] ( (1 + x)^(n+3)/(1 - x)^(n-3) )^k for n, k >= 0.

1, 1, 0, 1, -4, -20, 1, 0, 28, 0, 1, 20, -84, -220, 924, 1, 64, 924, 0, 1820, 0, 1, 140, 12012, 48620, 16796, -15504, -48620, 1, 256, 60060, 2621440, 2704156, 0, 134596, 0, 1, 420, 204204, 29745716, 608435100, 155117520, -3801900, -1184040, 2704156, 1, 640, 554268, 187432960, 15628090140, 146028888064, 9075135300, 0, 10518300, 0

Offset: 0

Author

Peter Bala, Aug 07 2023

Keywords

• tabl
• sign
• easy

Comments

Compare with A364303 and 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).

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.

It is known that A005810, the unsigned version of row 1, 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. We conjecture that each row sequence of the table satisfies the same supercongruences.

Examples

			Square array begins:
 n\k| 0    1       2          3             4                5
  - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  0 | 1    0     -20          0           924                0  ... see A066802
  1 | 1   -4      28       -220          1820           -15504  ... see A005810
  2 | 1    0     -84          0         16796                0
  3 | 1   20     924      48620       2704156        155117520  ... A066802
  4 | 1   64   12012    2621440     608435100     146028888064  ... A364520
  5 | 1  140   60060   29745716   15628090140    8480843582640  ... A211420
		

Links

• Peter Bala, Some integer ratios of factorials
• J. W. Bober, Factorial ratios, hypergeometric series, and a family of step functions, arXiv:0709.1977 [math.NT], 2007; J. London Math. Soc. (2) 79 2009, 422-444.
• Wikipedia, Hypergeometric function

Crossrefs

Cf. A066802 (row 3, also row 0 unsigned and without 0's), A005810 (row 1 unsigned), A364520 (row 4), A211420 (row 5).

Cf. A330843, A364303, A364518.

Programs

Maple

T(n,k) := add( binomial((n+3)*k, j)*binomial(n*k-j-1, 3*k-j), j = 0..3*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);

PARI

T(n,k) = sum(j = 0, 3*k, binomial((n+3)*k, j)*binomial(n*k-j-1, 3*k-j));
lista(nn) = for( n=0, nn, for (k=0, n, print1(T(n-k, k), ", "))); \\ Michel Marcus, Aug 13 2023

Formula

T(n,k) = Sum_{j = 0..3*k} binomial((n+3)*k, j)*binomial(n*k-j-1, 3*k-j).

For n >= 3, T(n,k) = binomial(n*k-1,3*k) * hypergeom([-(n+3)*k, -3*k], [1 - n*k], -1) = ((n+3)*k)!*((n-3)*k/2)!/(((n+3)*k/2)!*((n-3)*k)!*(3*k)!) by Kummer's Theorem.

The row generating functions are algebraic functions over the field of rational functions Q(x).