A295437 -id:A295437 - cp's OEIS frontend

A295431 a(n) = (12n)!n! / ((6n)!(4n)!(3*n)!).

1, 4620, 89237148, 2005604901300, 47913489552349980, 1183237138556438547120, 29836408028165719837829700, 763223193205837155576920270520, 19728995249931089572476730815356700, 514073874001824145407534840409364592528, 13479596359042448208364688886016106250225648

Offset: 0

Gheorghe Coserea, Nov 22 2017

nonn

From Peter Bala, Jan 24 2020: (Start)
a(p^k) == a(p^(k-1)) ( mod p^(3*k) ) for any prime p >= 5 and any positive integer k (write a(n) as C(12*n,6*n)*C(6*n,3*n)/C(4*n,n) and use Mestrovic, equation 39, p. 12).
More generally, for this sequence and the other integer factorial ratio sequences listed in the cross references, the congruences a(n*p^k) == a(n*p^(k-1)) ( mod p^(3*k) ) may hold for any prime p >= 5 and any positive integers n and k. (End)
a(n*p) == a(n) ( mod p^3 ) are proved for all such sequences in Section 5 of Zudilin's article. - Wadim Zudilin, Jul 30 2021

Gheorghe Coserea, Table of n, a(n) for n = 0..202
F. Beukers and Heckman, G., Monodromy for the hypergeometric function nFn-1", Inventiones mathematicae 95.2 (1989): 325-354.
Jonathan Bober, Factorial ratios, hypergeometric series, and a family of step functions, 2007, arXiv:0709.1977 [math.NT], 2007; J. London Math. Soc., Vol. 79, Issue 2 (2009), 422-444.
Gheorghe Coserea, Table with the parameters of the 52 sporadic integral factorial ratio sequences
R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.
F. Rodriguez-Villegas, Integral ratios of factorials and algebraic hypergeometric functions, arXiv:math/0701362 [math.NT], 2007.
Wadim Zudilin, Congruences for q-binomial coefficients, arXiv:1901.07843 [math.NT], 2019.

The 52 sporadic integral factorial ratio sequences:
Idx EntryID   u(r,s)         dFd-1
---+---------+--------------+-----------------------------------------------+
1   A295431   [12,1]         [1/12,5/12,7/12,11/12]
              [6,4,3]        [1/3,1/2,2/3]
2   A295432   [12,3,2]       [1/12,5/12,7/12,11/12]
              [6,6,4,1]      [1/6,1/2,5/6]
3   A295433   [12,1]         [1/12,1/6,5/12,7/12,5/6,11/12]
              [8,3,2]        [1/8,3/8,1/2,5/8,7/8]
4   A295434   [12,3]         [1/12,1/3,5/12,7/12,2/3,11/12]
              [8,6,1]        [1/8,3/8,1/2,5/8,7/8]
5   A295435   [12,3]         [1/12,1/3,5/12,7/12,2/3,11/12]
              [6,5,4]        [1/5,2/5,1/2,3/5,4/5]
6   A295436   [12,5]         [1/12,1/6,5/12,7/12,5/6,11/12]
              [10,4,3]       [1/10,3/10,1/2,7/10,9/10]
7   A295437   [18,1]         [1/18,5/18,7/18,11/18,13/18,17/18]
              [9,6,4]        [1/4,1/3,1/2,2/3,3/4]
8   A295438   [9,2]          [1/9,2/9,4/9,5/9,7/9,8/9]
              [6,4,1]        [1/6,1/4,1/2,3/4,5/6]
9   A295439   [9,4]          [1/9,2/9,4/9,5/9,7/9,8/9]
              [8,3,2]        [1/8,3/8,1/2,5/8,7/8]
10  A295440   [18,4,3]       [1/18,5/18,7/18,11/18,13/18,17/18]
              [9,8,6,2]      [1/8,3/8,1/2,5/8,7/8]
11  A295441   [9,1]          [1/9,2/9,4/9,5/9,7/9,8/9]
              [5,3,2]        [1/5,2/5,1/2,3/5,4/5]
12  A295442   [18,5,3]       [1/18,5/18,7/18,11/18,13/18,17/18]
              [10,9,6,1]     [1/10,3/10,1/2,7/10,9/10]
13  A295443   [18,4]         [1/18,5/18,7/18,1/2,11/18,13/18,17/18]
              [12,9,1]       [1/12,1/3,5/12,7/12,2/3,11/12]
14  A295444   [12,2]         [1/12,1/6,5/12,1/2,7/12,5/6,11/12]
              [9,4,1]        [1/9,2/9,4/9,5/9,7/9,8/9]
15  A295445   [18,2]         [1/18,5/18,7/18,1/2,11/18,13/18,17/18]
              [9,6,5]        [1/5,1/3,2/5,3/5,2/3,4/5]
16  A295446   [10,6]         [1/10,1/6,3/10,1/2,7/10,5/6,9/10]
              [9,5,2]        [1/9,2/9,4/9,5/9,7/9,8/9]
17  A295447   [14,3]         [1/14,3/14,5/14,1/2,9/14,11/14,13/14]
              [9,7,1]        [1/9,2/9,4/9,5/9,7/9,8/9]
18  A295448   [18,3,2]       [1/18,5/18,7/18,1/2,11/18,13/18,17/18]
              [9,7,6,1]      [1/7,2/7,3/7,4/7,5/7,6/7]
19  A295449   [12,2]         [1/12,1/6,5/12,1/2,7/12,5/6,11/12]
              [7,4,3]        [1/7,2/7,3/7,4/7,5/7,6/7]
20  A295450   [14,6,4]       [1/14,3/14,5/14,1/2,9/14,11/14,13/14]
              [12,7,3,2]     [1/12,1/3,5/12,7/12,2/3,11/12]
21  A295451   [14,1]         [1/14,3/14,5/14,1/2,9/14,11/14,13/14]
              [7,5,3]        [1/5,1/3,2/5,3/5,2/3,4/5]
22  A295452   [10,6,1]       [1/10,1/6,3/10,1/2,7/10,5/6,9/10]
              [7,5,3,2]      [1/7,2/7,3/7,4/7,5/7,6/7]
23  A295453   [15,1]         [1/15,2/15,4/15,7/15,8/15,11/15,13/15,14/15]
              [9,5,2]        [1/9,2/9,4/9,1/2,5/9,7/9,8/9]
24  A295454   [30,9,5]       [1/30,7/30,11/30,13/30,17/30,19/30,23/30,29/30]
              [18,15,10,1]   [1/18,5/18,7/18,1/2,11/18,13/18,17/18]
25  A295455   [15,4]         [1/15,2/15,4/15,7/15,8/15,11/15,13/15,14/15]
              [12,5,2]       [1/12,1/6,5/12,1/2,7/12,5/6,11/12]
26  A295456   [30,5,4]       [1/30,7/30,11/30,13/30,17/30,19/30,23/30,29/30]
              [15,12,10,2]   [1/12,1/3,5/12,1/2,7/12,2/3,11/12]
27  A295457   [15,4]         [1/15,2/15,4/15,7/15,8/15,11/15,13/15,14/15]
              [8,6,5]        [1/8,1/6,3/8,1/2,5/8,5/6,7/8]
28  A295458   [30,5,4]       [1/30,7/30,11/30,13/30,17/30,19/30,23/30,29/30]
              [15,10,8,6]    [1/8,1/3,3/8,1/2,5/8,2/3,7/8]
29  A295459   [15,2]         [1/15,2/15,4/15,7/15,8/15,11/15,13/15,14/15]
              [10,4,3]       [1/10,1/4,3/10,1/2,7/10,3/4,9/10]
30  A295460   [30,3,2]       [1/30,7/30,11/30,13/30,17/30,19/30,23/30,29/30]
              [15,10,6,4]    [1/5,1/4,2/5,1/2,3/5,3/4,4/5]
31  A211417   [30,1]         [1/30,7/30,11/30,13/30,17/30,19/30,23/30,29/30]
              [15,10,6]      [1/5,1/3,2/5,1/2,3/5,2/3,4/5]
32  A295462   [15,2]         [1/15,2/15,4/15,7/15,8/15,11/15,13/15,14/15]
              [10,6,1]       [1/10,1/6,3/10,1/2,7/10,5/6,9/10]
33  A295463   [15,7]         [1/15,2/15,4/15,7/15,8/15,11/15,13/15,14/15]
              [14,5,3]       [1/14,3/14,5/14,1/2,9/14,11/14,13/14]
34  A295464   [30,5,3]       [1/30,7/30,11/30,13/30,17/30,19/30,23/30,29/30]
              [15,10,7,6]    [1/7,2/7,3/7,1/2,4/7,5/7,6/7]
35  A295465   [30,5,3]       [1/30,7/30,11/30,13/30,17/30,19/30,23/30,29/30]
              [15,12,10,1]   [1/12,1/4,5/12,1/2,7/12,3/4,11/12]
36  A295466   [15,6,1]       [1/15,2/15,4/15,7/15,8/15,11/15,13/15,14/15]
              [12,5,3,2]     [1/12,1/4,5/12,1/2,7/12,3/4,11/12]
37  A295467   [15,1]         [1/15,2/15,4/15,7/15,8/15,11/15,13/15,14/15]
              [8,5,3]        [1/8,1/4,3/8,1/2,5/8,3/4,7/8]
38  A295468   [30,5,3,2]     [1/30,7/30,11/30,13/30,17/30,19/30,23/30,29/30]
              [15,10,8,6,1]  [1/8,1/4,3/8,1/2,5/8,3/4,7/8]
39  A295469   [20,3]         [1/20,3/20,7/20,9/20,11/20,13/20,17/20,19/20]
              [12,10,1]      [1/12,1/6,5/12,1/2,7/12,5/6,11/12]
40  A295470   [20,6,1]       [1/20,3/20,7/20,9/20,11/20,13/20,17/20,19/20]
              [12,10,3,2]    [1/12,1/3,5/12,1/2,7/12,2/3,11/12]
41  A295471   [20,1]         [1/20,3/20,7/20,9/20,11/20,13/20,17/20,19/20]
              [10,8,3]       [1/8,1/3,3/8,1/2,5/8,2/3,7/8]
42  A295472   [20,3,2]       [1/20,3/20,7/20,9/20,11/20,13/20,17/20,19/20]
              [10,8,6,1]     [1/8,1/6,3/8,1/2,5/8,5/6,7/8]
43  A061164   [20,1]         [1/20,3/20,7/20,9/20,11/20,13/20,17/20,19/20]
              [10,7,4]       [1/7,2/7,3/7,1/2,4/7,5/7,6/7]
44  A295474   [20,7,2]       [1/20,3/20,7/20,9/20,11/20,13/20,17/20,19/20]
              [14,10,4,1]    [1/14,3/14,5/14,1/2,9/14,11/14,13/14]
45  A295475   [20,3]         [1/20,3/20,7/20,9/20,11/20,13/20,17/20,19/20]
              [10,9,4]       [1/9,2/9,4/9,1/2,5/9,7/9,8/9]
46  A295476   [20,9,6]       [1/20,3/20,7/20,9/20,11/20,13/20,17/20,19/20]
              [18,10,4,3]    [1/18,5/18,7/18,1/2,11/18,13/18,17/18]
47  A295477   [24,1]         [1/24,5/24,7/24,11/24,13/24,17/24,19/24,23/24]
              [12,8,5]       [1/5,1/4,2/5,1/2,3/5,3/4,4/5]
48  A295478   [24,5,2]       [1/24,5/24,7/24,11/24,13/24,17/24,19/24,23/24]
              [12,10,8,1]    [1/10,1/4,3/10,1/2,7/10,3/4,9/10]
49  A295479   [24,4,1]       [1/24,5/24,7/24,11/24,13/24,17/24,19/24,23/24]
              [12,8,7,2]     [1/7,2/7,3/7,1/2,4/7,5/7,6/7]
50  A295480   [24,7,4]       [1/24,5/24,7/24,11/24,13/24,17/24,19/24,23/24]
              [14,12,8,1]    [1/14,3/14,5/14,1/2,9/14,11/14,13/14]
51  A295481   [24,4,3]       [1/24,5/24,7/24,11/24,13/24,17/24,19/24,23/24]
              [12,9,8,2]     [1/9,2/9,4/9,1/2,5/9,7/9,8/9]
52  A295482   [24,9,6,4]     [1/24,5/24,7/24,11/24,13/24,17/24,19/24,23/24]
              [18,12,8,3,2]  [1/18,5/18,7/18,1/2,11/18,13/18,17/18]
Cf. A304126.

seq((12*n)!*n!/((6*n)!*(4*n)!*(3*n)!),n=0..10); # Karol A. Penson, May 08 2018

Table[((12n)!n!)/((6n)!(4n)!(3n)!),{n,0,20}] (* Harvey P. Dale, Sep 14 2019 *)

r=[12,1]; s=[6,4,3];
p=[1/12,5/12,7/12,11/12]; q=[1/3,1/2,2/3];
C(r,s) = prod(k=1, #r, r[k]^r[k])/prod(k=1, #s, s[k]^s[k]);
u(r, s, N=20) = {
  my(f=(v,n)->prod(k=1, #v, (v[k]*n)!));
  apply(n->f(r,n)/f(s,n), [0..N-1]);
};
u(r,s,11)
\\ test 1:
\\ system("wget http://www.jjj.de/pari/hypergeom.gpi");
read("hypergeom.gpi");
N=200; x='x+O('x^N); u(r,s,N) == Vec(hypergeom(p, q, C(r,s)*x, N))
\\ test 2: check consistency of all parameters
system("wget https://oeis.org/A295431/a295431.txt");
N=200; x='x+O('x^N); w = read("a295431.txt");
52==vecsum(vector(#w, n, u(w[n][1], w[n][2], N) == Vec(hypergeom(w[n][3], w[n][4], C(w[n][1], w[n][2])*x, N))))

G.f.: hypergeom([1/12, 5/12, 7/12, 11/12], [1/3, 1/2, 2/3], 27648*x).
From Karol A. Penson, May 08 2018 (Start):
Asymptotics: a(n) ~ (2^n)^10*(3^n)^3*sqrt(3/n)*(2592*n^2+72*n+1)/(15552*n^2*sqrt(Pi)), for n->infinity.
Integral representation as the n-th moment of the positive function V(x) on x = (0, 27648), i.e. in Maple notation: a(n) = int(x^n*V(x), x = 0..27648), n=0,1..., where V(x) = 3^(3/4)*sqrt(2)*hypergeom([1/12, 5/12, 7/12, 3/4], [1/6, 1/2, 2/3], (1/27648)*x)*GAMMA(3/4)/(36*sqrt(Pi)*x^(11/12)*GAMMA(2/3)*GAMMA(7/12))+3^(1/4)*sqrt(2)*cos(5*Pi*(1/12))*GAMMA(2/3)*csc((1/12)*Pi)*GAMMA(3/4)*hypergeom([5/12, 3/4, 11/12, 13/12], [1/2, 5/6, 4/3], (1/27648)*x)/(4608*Pi^(3/2)*GAMMA(11/12)*x^(7/12))+3^(1/4)*cos(5*Pi*(1/12))*GAMMA(11/12)*hypergeom([7/12, 11/12, 13/12, 5/4], [2/3, 7/6, 3/2], (1/27648)*x)/(6912*sqrt(Pi)*GAMMA(2/3)*GAMMA(3/4)*x^(5/12))+7*3^(3/4)*sin(5*Pi*(1/12))*GAMMA(2/3)*GAMMA(7/12)*hypergeom([11/12, 5/4, 17/12, 19/12], [4/3, 3/2, 11/6], (1/27648)*x)/(2654208*Pi^(3/2)*GAMMA(3/4)*x^(1/12)). The function V(x) is singular at both edges of its support and is U-shaped. The function V(x) is unique as it is the solution of the Hausdorff moment problem. (End)
D-finite with recurrence: n*(3*n-1)*(2*n-1)*(3*n-2)*a(n) -24*(12*n-11)*(12*n-1)*(12*n-5)*(12*n-7)*a(n-1)=0. - R. J. Mathar, Jan 27 2020

A347858 a(n) = (9n)!/((3n)!(2n)!) * (n/2)!/(9*n/2)!.

Original entry on oeis.org

1, 512, 1021020, 2399141888, 6016814703900, 15626259253952512, 41477110789150966020, 111745115394167694950400, 304331361887290342345862940, 835666006020766806513664655360, 2309513382640863232775760738593520, 6416034331756986890806503962421755904

Offset: 0

Peter Bala, Sep 19 2021

Fractional factorials are defined using the Gamma function; for example, (9*n/2)! := Gamma(1 + 9*n/2).

The sequence defined by u(n) = (18*n)!*(n)!/((4*n)!*(6*n)!*(9*n)!) is one of the 52 sporadic integral factorial ratio sequences of height 1 found by V. I. Vasyunin (see Bober, Table 2, Entry 7). See A295437. Here we are essentially considering the sequence (u(n/2))n>=0. The sequence is conjectured to be integral.

			Congruence: a(11) - a(1) = 6416034331756986890806503962421755904 - 512 = (2^9)*(11^3)*7207*1306363809854375553476366323 == 0 (mod 11^3).

J. W. Bober, Factorial ratios, hypergeometric series, and a family of step functions, arXiv:0709.1977 [math.NT], 2007; J. London Math. Soc., Vol. 79, Issue 2 (2009), 422-444.
F. Rodriguez-Villegas, Integral ratios of factorials and algebraic hypergeometric functions, arXiv:math/0701362 [math.NT], 2007.

Cf. A295437, A347854, A347855, A347856, A347857.

seq( (9*n)!/((3*n)!*(2*n)!) * GAMMA(1+n/2)/GAMMA(1+9*n/2), n = 0..11);

from math import factorial
from sympy import factorial2
def A347858(n): return int((factorial(9*n)*factorial2(n)<<(n<<2))//(factorial(3*n)*factorial(n<<1)*factorial2(9*n))) # Chai Wah Wu, Aug 10 2023

a(n) = binomial(9*n,3*n)*binomial(6*n,2*n)/binomial(9*n/2,4*n).
a(2*n) = A295437(n).
a(2*n) = 72*(18*n-1)*(18*n-5)*(18*n-7)*(18*n-11)*(18*n-13)*(18*n-17)/(n*(2*n-1)*(3*n-1)*(3*n-2)*(4*n-1)*(4*n-3))*a(2*n-2);
a(2*n+1) = 18432*(81*n^2-1)*(81*n^2-4)*(81*n^2-16)/(n*(2*n+1)*(16*n^2-1)*(36*n^2-1))*a(2*n-1).
Asymptotics: a(n) ~ 1/(2*sqrt(3*Pi*n)) * 2916^n as n -> infinity.
O.g.f.: A(x) = hypergeom([1/18, 5/18, 7/18, 11/18, 13/18, 17/18], [1/4, 1/3, 1/2, 2/3, 3/4], (2^4)*(3^12)*x^2) + 512*x*hypergeom([5/9, 7/9, 8/9, 10/9, 11/9, 13/9], [3/4, 5/6, 7/6, 5/4, 3/2], (2^4)*(3^12)*x^2) is conjectured to be algebraic over Q(x).
Conjectural congruences: a(n*p^k) == a(n*p^(k-1)) ( mod p^(3*k) ) for prime p >= 5 and positive integers n and k.

A364172 a(n) = (6n)!(n/3)!/((3n)!(2n)!(4*n/3)!).

Original entry on oeis.org

1, 45, 6237, 1021020, 178719453, 32427545670, 6016814703900, 1133540594837892, 215925912619400925, 41477110789150966020, 8019784929635201045862, 1558875476359831844951100, 304331361887290342345862940, 59629409730107012112361325820

Offset: 0

Peter Bala, Jul 12 2023

Fractional factorials are defined in terms of the gamma function; for example, (n/3)! := Gamma(n/3 + 1).
Given two sequences of numbers 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. For a list of the 52 sporadic integral factorial ratio sequences see A295431.
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 rational numbers.
A295437, defined by A295437(n) = (18*n)!*n! / ((9*n)!*(6*n)!*(4*n)!) is one of the 52 sporadic integral factorial ratio sequences of height 1 found by V. I. Vasyunin (see Bober, Table 2, Entry 7). Here we are essentially considering the sequence {A295437(n/3) : n >= 0}. This sequence is only conjecturally an integer sequence.
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.

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., 79, Issue 2, (2009), 422-444.

Cf. A276100, A276101, A276102, A295431, A295437, A347854, A347855, A347856, A347857, A347858, A364173 - A364185.

seq( simplify((6*n)!*(n/3)!/((3*n)!*(2*n)!*(4*n/3)!)), n = 0..15);

Table[Product[36*(6*k - 5)*(6*k - 1)/(k*(3*k + n)), {k, 1, n}], {n, 0, 20}] (*  Vaclav Kotesovec, Jul 13 2023 *)

a(n) ~ 2^(4*n/3 - 3/2) * 3^(4*n) / sqrt(Pi*n). - Vaclav Kotesovec, Jul 13 2023

a(n) = 5832*(6*n - 1)*(6*n - 5)*(6*n - 7)*(6*n - 11)*(6*n - 13)*(6*n - 17)/(n*(n - 1)*(n - 2)*(2*n - 3)*(4*n - 3)*(4*n - 9))*a(n-3) for n >= 3 with a(0) = 1, a(1) = 45 and a(2) = 6237.

cp's OEIS Frontend

A295431 a(n) = (12n)!n! / ((6n)!(4n)!(3*n)!).

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A347858 a(n) = (9n)!/((3n)!(2n)!) * (n/2)!/(9*n/2)!.

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A364172 a(n) = (6n)!(n/3)!/((3n)!(2n)!(4*n/3)!).

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cp's OEIS Frontend

A295431 a(n) = (12*n)!*n! / ((6*n)!*(4*n)!*(3*n)!).

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Author

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Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A347858 a(n) = (9*n)!/((3*n)!*(2*n)!) * (n/2)!/(9*n/2)!.

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Author

Keywords

Comments

Examples

Links

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Programs

Maple

Python

Formula

A364172 a(n) = (6*n)!*(n/3)!/((3*n)!*(2*n)!*(4*n/3)!).

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Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A295431 a(n) = (12n)!n! / ((6n)!(4n)!(3*n)!).

A347858 a(n) = (9n)!/((3n)!(2n)!) * (n/2)!/(9*n/2)!.

A364172 a(n) = (6n)!(n/3)!/((3n)!(2n)!(4*n/3)!).