OFFSET
0,2
COMMENTS
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
LINKS
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.
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.
FORMULA
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
MAPLE
seq((12*n)!*n!/((6*n)!*(4*n)!*(3*n)!), n=0..10); # Karol A. Penson, May 08 2018
MATHEMATICA
Table[((12n)!n!)/((6n)!(4n)!(3n)!), {n, 0, 20}] (* Harvey P. Dale, Sep 14 2019 *)
PROG
(PARI)
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))))
CROSSREFS
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.
KEYWORD
nonn
AUTHOR
Gheorghe Coserea, Nov 22 2017
STATUS
approved