A364514 -id:A364514

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A364513

Square array read by ascending antidiagonals: T(n,k) = [x^k] (1 - x)^(2*k) * Legendre_P(n*k-1, (1 + x)/(1 - x)) for n, k >= 0.

+0
6

1, 1, -2, 1, -2, 6, 1, 0, 0, -20, 1, 4, 0, 16, 70, 1, 10, 126, 0, 0, -252, 1, 18, 594, 4900, 0, -252, 924, 1, 28, 1716, 44200, 209950, 0, 0, -3432, 1, 40, 3900, 205920, 3640210, 9513504, 0, 4800, 12870, 1, 54, 7650, 685216, 27386100, 317678760, 447103440, 0, 0, -48620, 1, 70, 13566, 1847560, 133501500, 3861534768, 28782923400, 21558808128, 0, -100100, 184756

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Compare with A364303.

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). Soundararajan gives many examples of two-parameter families of integral factorial ratio sequences of height 2.

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

LINKS

Table of n, a(n) for n=0..65.

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.

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 = 0..k} binomial(n*k-1, k-i)^2 * binomial((n-2)*k+i-2, i).

T(n,1) = 1 for all n and for n >= 2 and k >= 1, T(n,k) = binomial((k*n-1), k)^2 * hypergeom([a, b, b], [1 + a - b, 1 + a - b], 1), where a = (n - 2)*k - 1 and b = -k.

For n >= 3 and k >= 1, T(n,k) = ((n*k - 1))! * ( ((n+2)*k - 1)/2 )! * ( ((n-2)*k - 1)/2 )! / ( k!^2 * ((n-2)*k - 1)! * ((n*k - 1)/2)!^2 ) by Dixon's 3F2 summation theorem, where fractional factorials are defined in terms of the gamma function.

For n >= 3 and k >= 1, T(n,k) = (n-2)/n * ((n+2)*k)!*(n*k/2)!^2 / ( ((n+2)*k/2)! * (n*k)! * ((n-2)*k/2)! * k!^2 ).

The central binomial numbers A000984, row 1 unsigned, satisfy the supercongruences A000984(n*p^r) == A000984(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.

EXAMPLE

Square array begins:

n\k| 0 1 2 3 4 5

- + - - - - - - - - - - - - - - - - - - - - - - - - - - -

0 | 1 -2 6 -20 70 -252 ... (see A000984)

1 | 1 -2 0 16 0 -252 ... A364514

2 | 1 0 0 0 0 0

3 | 1 4 126 4900 209950 9513504 ... (1/3)*A352651

4 | 1 10 594 44200 3640210 317678760 ... A364515

5 | 1 18 1716 205920 27386100 3861534768 ... (3/5)*A352652

6 | 1 28 3900 685216 133501500 27583083528 ... A364516

7 | 1 40 7650 1847560 494944450 140625140040 ... A364517

MAPLE

T(n, k) := coeff(series( (1 - x)^(2*k) * LegendreP(n*k-1, (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

Cf. A000984 (row 0 unsigned), A276098, A295431, A352651 (3*row 3), A352652 ((5/3)*row 5), A364303, A364506, A364509, A364514 (row 1), A364515 (row 4), A364516 (row 6), A364517 (row 7).

KEYWORD

sign,tabl,easy

AUTHOR

Peter Bala, Jul 31 2023

STATUS

approved

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