A364303 -id:A364303 - cp's OEIS frontend

A245086 Central values of the n-th discrete Chebyshev polynomials of order 2n.

1, 0, -6, 0, 90, 0, -1680, 0, 34650, 0, -756756, 0, 17153136, 0, -399072960, 0, 9465511770, 0, -227873431500, 0, 5550996791340, 0, -136526995463040, 0, 3384731762521200, 0, -84478098072866400, 0, 2120572665910728000, 0, -53494979785374631680, 0

Offset: 0

Nikita Gogin, Jul 11 2014

In the general case the n-th discrete Chebyshev polynomial of order N is D(N,n;x) = Sum_{i = 0..n} (-1)^i*C(n,i)*C(N-x,n-i)*C(x,i). For N = 2*n , x = n, one gets a(n) = D(2n,n;n) = Sum_{i = 0..n} (-1)^i*C(n,i)^3 that equals (due to Dixon's formula) 0 for odd n and (-1)^m*(3m)!/(m!)^3 for n = 2*m. (Riordan, 1968) So, a(2*m) = (-1)^m*A006480(m).

John Riordan, Combinatorial Identities, John Willey&Sons Inc., 1968.

Nikita Gogin and Mika Hirvensalo, On the Generating Function of Discrete Chebyshev Polynomials, TUCS Technical Reports 819, Turku Centre for Computer Science, 2007.
Nikita Gogin and Mika Hirvensalo, Recurrent Constructions of the MacWilliams and Chebyshev Matrices, Fundamenta Informaticae, vol.116, issue 1-4, January 2012, pages 93-110.
H. W. Gould, Tables of Combinatorial Identities, Edited by J. Quaintance.
Darij Grinberg, Introduction to Modern Algebra (UMN Spring 2019 Math 4281 Notes), University of Minnesota (2019).
Y. Sun and X. Wang, A new proof of a curious identity, Math Gazette, Vol. 91, No. 520 (Mar 2007) 105-107.
Wikipedia, Dixon's identity

Cf. A000172, A002894, A005260, A006480, A103882, A183204, A273630, A364303.

Table[Coefficient[Simplify[JacobiP[n,0,-(2*n+1),(1+t^2)/(1-t^2)]*(1-t^2)^n],t,n],{n,0,20}]

from math import factorial
def A245086(n): return 0 if n&1 else (-1 if (m:=n>>1)&1 else 1)*factorial(3*m)//factorial(m)**3 # Chai Wah Wu, Oct 04 2022

a(n) is a coefficient at t^n in (1-t^2)^n*P(0,-(2*n+1);n;(1+t^2)/(1-t^2)), where P(a,b;k;x) is the k-th Jacobi polynomial (Gogin and Hirvensalo, 2007).
G.f.: Hypergeometric2F1[1/3,2/3,1,-27*x^2].
a(2*m+1) = 0, a(2*m) = (-1)^m*A006480(m).
From Peter Bala, Aug 04 2016: (Start)
a(n) = Sum_{k = 0..n} (-1)^k*binomial(n,k)*binomial(2*n - k,n)*binomial(n + k,n) (Sun and Wang).
a(n) = Sum_{k = 0..n} (-1)^(n + k)*binomial(n + k, n - k)*binomial(2*k, k)*binomial(2*n - k, n) (Gould, Vol.5, 9.23).
a(n) = -1/(n + 1)^3 * A273630(n+1). (End)
From Peter Bala, Mar 22 2022: (Start)
a(n) = - (3*(3*n-2)*(3*n-4)/n^2)*a(n-2).
a(n) = [x^n] (1 - x)^(2*n) * P(n,(1 + x)/(1 - x)), where P(n,x) denotes the n-th Legendre polynomial. Compare with A002894(n) = binomial(2*n,n)^2 = [x^n] (1 - x)^(2*n) * P(2*n,(1 + x)/(1 - x)). Cf. A103882. (End)
From Peter Bala, Jul 23 2023: (Start)
a(n) = [x^n] G(x)^(3*n), where the power series G(x) = 1 - x^2 + 2*x^4 - 14*x^6 + 127*x^8 - 1364*x^10 + ... appears to have integer coefficients.
exp(Sum_{n >= 1} a(n)*x^n/n) = F(x)^3, where the power series F(x) = 1 - x^2  + 8*x^4 - 101*x^6 + 1569*x^8 - 27445*x^10 + ..., appears to have integer coefficients. See A229452.
Row 1 of A364303. (End)
a(n) = Sum_{k = 0..n} (-1)^(n-k) * binomial(n+k, k)^2 * binomial(3*n+1, n-k). Cf. A183204.- Peter Bala, Sep 20 2024

A364509 Square array read by ascending antidiagonals: T(n,k) = (2k)!/k!^2 ( (2nk)! * ((n + 2)k)! )/( (nk)! * ((n + 1)*k)!^2 ) for n, k > = 0.

Original entry on oeis.org

1, 1, 4, 1, 6, 36, 1, 16, 90, 400, 1, 50, 784, 1680, 4900, 1, 168, 8910, 48400, 34650, 63504, 1, 588, 113256, 2011100, 3312400, 756756, 853776, 1, 2112, 1528436, 96993024, 503909070, 240374016, 17153136, 11778624, 1, 7722, 21395520, 5056527000, 92279796840, 133954543800, 18116083216

Offset: 0

Peter Bala, Jul 28 2023

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 squares of 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 [added Oct 11 2024: follows from Meštrović, Section 6, equation 39, since T(n, k) = binomial(2*k, k) * binomial(2*n*k, n*k) * binomial((n+2)*k, k)/binomial((n+1)*k, k)].

			 Square array begins:
 n\k|  0    1        2           3               4                  5
  - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  0 |  1    4       36         400            4900              63504 ... A002894
  1 |  1    6       90        1680           34650             756756 ... A006480
  2 |  1   16      784       48400         3312400          240374016 ... A364510
  3 |  1   50     8910     2011100       503909070       133954543800 ... A364511
  4 |  1  168   113256    96993024     92279796840     93172920645168 ...
  5 |  1  588  1528436  5056527000  18592935952500  72567511917065088 ...

Winston de Greef, Table of n, a(n) for n = 0..3240 (80 antidiagonals)
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.
Romeo Meštrović, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2012), arXiv:1111.3057 [math.NT], (2011).
K. Soundararajan, Integral factorial ratios: irreducible examples with height larger than 1, Phil. Trans. Royal Soc., A378: 2018044, 2019.
Wikipedia, Dixon's identity

A002894 (row 0), A006480 (row 1), A364510 (row 3), A364511 (row 4).

Cf. A364303, A364506.

 # display as a square array
T(n,k) := (2*k)!/k!^2 * ( (2*n*k)! * ((n + 2)*k)! )/( (n*k)! * ((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);

T(n,k) = (2*k)!/k!^2 * ( (2*n*k)! * ((n + 2)*k)! )/( (n*k)! * ((n + 1)*k)!^2 ) \\ Winston de Greef, Oct 05 2023

T(n,k) = Sum_{i = -k..k} (-1)^i * binomial(2*k, k+i)^2 * binomial(2*n*k, n*k+i) (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, -2*k, -(n+1)*k], [1, 1 + (n-1)*k], 1) = (2*k)!/k!^2 * ( (2*n*k)! * ((n + 2)*k)! )/( (n*k)! * ((n + 1)*k)!^2 ) by Dixon's 3F2 summation theorem.
T(n,k) = (-1)^(n*k) * [x^((n+1)*k)] ( (1 - x)^(2*(n+1)*k) * Legendre_P(2*k, (1 + x)/(1 - x)) ). - Peter Bala, Aug 14 2023

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

Original entry on oeis.org

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

Peter Bala, Jul 27 2023

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.

			 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  ...

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

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

# 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);

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

A364513 Square array read by ascending antidiagonals: T(n,k) = [x^k] (1 - x)^(2k) Legendre_P(n*k-1, (1 + x)/(1 - x)) for n, k >= 0.

Original entry on oeis.org

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

Offset: 0

Peter Bala, Jul 31 2023

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.

			 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

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

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).

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);

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.

A364304 a(n) = (7n)!(9n/2)!(5n/2)!/((5n)!(7n/2)!^2*n!^2).

Original entry on oeis.org

1, 54, 10296, 2484000, 665091000, 188907932304, 55737530929080, 16888537352985408, 5218680924762089400, 1637124203403474142500, 519752205290081232622296, 166620892958456148158454144, 53846423260084127389865311800

Offset: 0

Peter Bala, Jul 21 2023

Fractional factorials are defined in terms of the gamma function; for example, (9*n/2)! = gamma(1 + 9*n/2).

Paolo Xausa, Table of n, a(n) for n = 0..300

Row 7 of A364303.

Cf. A275652, A275654.

seq( simplify((7*n)!*(9*n/2)!*(5*n/2)!/((5*n)!*(7*n/2)!^2*n!^2)), n = 0..12);

A364304[n_]:=(7n)!(9n/2)!(5n/2)!/((5n)!(7n/2)!^2n!^2);Array[A364304,15,0] (* Paolo Xausa, Oct 06 2023 *)

from math import factorial
from sympy import factorial2
def A364304(n): return int(factorial(7*n)*factorial2(9*n)*factorial2(5*n)//factorial(5*n)//factorial2(7*n)**2//factorial(n)**2) # Chai Wah Wu, Aug 10 2023

a(n) = Sum_{k = 0..n} binomial(7*n, n - k)^2 * binomial(5*n + k - 1, k).
a(n) = Sum_{k = 0..n} (-1)^(n+k) * binomial(7*n + k, 7*n - k)*binomial(2*k, k)*binomial(2*n - k, n).
Conjecture: a(n) =  Sum_{k = 0..7*n} (-1)^k * binomial(7*n + k, 7*n - k)* binomial(2*k, k)*binomial(2*n - k, n)
a(n) = [x^n] (1 - x)^(2*n) * P(7*n, (1 + x)/(1 - x)), where P(n,x) denotes the n-th Legendre polynomial.
a(n) = [x^n] G(x)^(9*n), where G(x) = 1 + 6*x + 266*x^2 + 27104*x^3 + 3726380*x^4 + 600232416*x^5 + 106662768380*x^6 + ... appears to have integer coefficients.
exp( Sum_{n > = 1} a(n)*x^n/n ) = F(x)^9, where F(x) = 1 + 6*x + 590*x^2 + 95468*x^3 + 19200692*x^4 + 4364084760*x^5 + 1072849548644*x^6 + ... appears to have integer coefficients.
a(p) == a(1) (mod p^3).
Conjecture: the supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) hold for all primes p >= 5 and all positive integers n and r.
P-recursive: a(n) =  9*(9*n-2)*(9*n-4)*(9*n-6)*(9*n-8)*(9*n-10)*(9*n-12)*(9*n-14)*(9*n-16)*(7*n-1)*(7*n-3)*(7*n-5)*(7*n-9)*(7*n-11)*(7*n-13)/((7*n-2)*(7*n-4)*(7*n-6)*(7*n-8)*(7*n-10)*(7*n-12)*(5*n-1)*(5*n-3)*(5*n-5)*(5*n-7)*(5*n-9)*n^2*(n-1)) * a(n-2) with a(0) = 1 and a(1) = 54.
a(n) ~ c^n * 3*sqrt(7)/(14*Pi*n), where c = (3^9)/(5^3) * sqrt(5).

A364305 a(n) = (8n)!(5n)!(3n)! / ( (6n)!(4n)!^2*n!^2 ).

Original entry on oeis.org

1, 70, 17550, 5567380, 1960044750, 732012601320, 283986961467300, 113142133870180800, 45969979122504907470, 18961650930856541865100, 7915377251895103264073800, 3336455614603881320759754000, 1417729131896719482585245182500, 606517077508008639090614765297280

Offset: 0

Peter Bala, Jul 21 2023

Paolo Xausa, Table of n, a(n) for n = 0..300
R. Meštrović, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.

Row 8 of A364303.

Cf. A001449, A211421.

seq( (8*n)!*(5*n)!*(3*n)! / ( (6*n)!*(4*n)!^2*n!^2 ), n = 0..13);

A364305[n_]:=(8n)!(5n)!(3n)!/((6n)!(4n)!^2n!^2);Array[A364305,15,0] (* Paolo Xausa, Oct 06 2023 *)

a(n) = Sum_{k = 0..n} binomial(8*n, n-k)^2 * binomial(6*n+k-1, k).
a(n) = [x^n] (1 - x)^(2*n) * P(8*n,(1 + x)/(1 - x)), where P(n,x) denotes the n-th Legendre polynomial.
a(n) = (5/12)*(5*n-1)*(5*n-2)*(5*n-3)*(5*n-4)*(8*n-1)*(8*n-3)*(8*n-5)*(8*n-7)/((4*n-1)*(4*n-3)*(6*n-1)*(6*n-5)*n^2*(2*n-1)^2) * a(n-1) with a(0) = 1.
a(n) ~ c^n * sqrt(5)/(4*Pi*n), where c = (2^2)*(5^5)/(3^3).
a(n) = binomial(8*n,2*n)*binomial(5*n,n)*binomial(2*n,n)/binomial(4*n,n) = A001449(n) * A211421(n).
a(p) == a(1) (mod p^3) for all primes p >= 5.
Conjecture: the supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(3*k)) hold for all primes p >= 5 and all positive integers n and k [added 16 Oct 2024: the conjecture follows from Meštrović, equation 39, since a(n) = binomial(8*n, 2*n)*binomial(5*n, n)* binomial(2*n, n)/binomial(4*n, n)].
a(n) = [x^n] G(x)^(10*n), where the power series G(x) = 1 + 7*x + 412*x^2 + 55524*x^3 + 10088066*x^4 + 2146473322*x^5 + 503731865112*x^6 + ... appears to have integer coefficients.
exp(Sum_{n >= 1} a(n)*x^n/n) = F(x)^10, where the power series F(x) = 1 + 7*x + 902*x^2 + 191779*x^3 + 50706776*x^4 + 15153397742*x^5 + 4898289306180*x^6 + ... appears to have integer coefficients.
From Peter Bala, Oct 16 2024: (Start)
For integer r and positive integer s, define sequences u(n) = { [x^(s*n)] G(x)^(r*n) : n >= 0 } and v(n) = { [x^(s*n)] F(x)^(r*n) : n >= 0 }, with the power series F(x) and G(x) as defined above. We conjecture that both sequences {u(n)} and {v(n)} satisfy the above supercongruences mod p^(3*k). (End)

A364518 Square array read by ascending antidiagonals: T(n,k) = [x^(2*k)] ( (1 + x)^(n+2)/(1 - x)^(n-2) )^k for n, k >= 0.

Original entry on oeis.org

1, 1, -2, 1, 0, 6, 1, 6, -10, -20, 1, 16, 70, 0, 70, 1, 30, 630, 924, 198, -252, 1, 48, 2310, 28672, 12870, 0, 924, 1, 70, 6006, 204204, 1385670, 184756, -4420, -3432, 1, 96, 12870, 860160, 19122246, 69206016, 2704156, 0, 12870, 1, 126, 24310, 2704156, 130378950, 1848483780, 3528923580, 40116600, 104006, -48620

Offset: 0

Peter Bala, Aug 07 2023

Compare with A364303 and A364519.
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 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 the unsigned version of row 0 (the central binomial numbers A000984) and row 2 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.

			 Square array begins:
 n\k|  0   1      2        3           4             5
  - + - - - - - - - - - - - - - - - - - - - - - - - - -
  0 |  1  -2      6      -20          70          -252   ... see A000984
  1 |  1   0    -10        0         198             0   ... see A211419
  2 |  1   6     70      924       12870        184756   ... A001448
  3 |  1  16    630    28672     1385670      69206016   ... A091496
  4 |  1  30   2310   204204    19122246    1848483780   ... A061162
  5 |  1  48   6006   860160   130378950   20392706048   ... A276098
  6 |  1  70  12870  2704156   601080390  137846528820   ... A001448 bisected
  7 |  1  96  24310  7028736  2149374150  678057476096   ... A276099

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.
K. Soundararajan, Integral factorial ratios: irreducible examples with height larger than 1, Phil. Trans. Royal Soc., A378: 2018044, 2019.
Wikipedia, Dixon's identity
Wikipedia, Hypergeometric function

Cf. A000984 (row 0 unsigned), A211419 (row 1 unsigned without 0's), A001448 (row 2), A091496 (row 3), A061162 (row 4), A276098 (row 5), A001448 bisected (row 6), A276099 (row 7).

Cf. A330843, A364303, A364506, A364509, A364513, A364519.

T(n,k) = add( binomial((n+2)*k, j)*binomial(n*k-j-1, 2*k-j), j = 0..2*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);

T(n,k) = sum(j = 0, 2*k, binomial((n+2)*k, j)*binomial(n*k-j-1, 2*k-j));
lista(nn) = for( n=0, nn, for (k=0, n, print1(T(n-k, k), ", "))); \\ Michel Marcus, Aug 13 2023

T(n,k) = Sum_{j = 0..2*k} binomial((n+2)*k, j)*binomial(n*k-j-1, 2*k-j).
T(2,k) = binomial(4*k,2*k).
For n >= 3, T(n,k) = binomial(n*k-1,2*k) * hypergeom([-(n+2)*k, -2*k], [1 - n*k], -1) except when (n,k) = (3,1).
For n >= 2, T(n,k) = ((n+2)*k)!*((n-2)*k/2)!/(((n+2)*k/2)!*((n-2)*k)!*(2*k)!) by Kummer's Theorem.
T(n,k) = [x^k] (1 - x)^(2*k) * Chebyshev_T(n*k, (1 + x)/(1 - x)).
T(n,k) = Sum_{j = 0..k} binomial(2*n*k, 2*j)*binomial((n-1)*k-j-1, k-j).
For n >= 3, T(n,k) = binomial((n-1)*k-1,k) * hypergeom([-n*k, -k, -n*k + 1/2], [1 - (n-1)*k, 1/2], 1).
The row generating functions are algebraic functions over the field of rational functions Q(x).

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.

Original entry on oeis.org

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

Peter Bala, Aug 07 2023

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.

			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

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

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.

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);

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

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).

cp's OEIS Frontend

A245086 Central values of the n-th discrete Chebyshev polynomials of order 2n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Python

Formula

A364509 Square array read by ascending antidiagonals: T(n,k) = (2*k)!/k!^2 * ( (2*n*k)! * ((n + 2)*k)! )/( (n*k)! * ((n + 1)*k)!^2 ) for n, k > = 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

Formula

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 ).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A364304 a(n) = (7*n)!*(9*n/2)!*(5*n/2)!/((5*n)!*(7*n/2)!^2*n!^2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

A364305 a(n) = (8*n)!*(5*n)!*(3*n)! / ( (6*n)!*(4*n)!^2*n!^2 ).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A364518 Square array read by ascending antidiagonals: T(n,k) = [x^(2*k)] ( (1 + x)^(n+2)/(1 - x)^(n-2) )^k for n, k >= 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

A364509 Square array read by ascending antidiagonals: T(n,k) = (2k)!/k!^2 ( (2nk)! * ((n + 2)k)! )/( (nk)! * ((n + 1)*k)!^2 ) for n, k > = 0.

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

A364513 Square array read by ascending antidiagonals: T(n,k) = [x^k] (1 - x)^(2k) Legendre_P(n*k-1, (1 + x)/(1 - x)) for n, k >= 0.

A364304 a(n) = (7n)!(9n/2)!(5n/2)!/((5n)!(7n/2)!^2*n!^2).

A364305 a(n) = (8n)!(5n)!(3n)! / ( (6n)!(4n)!^2*n!^2 ).