%I #69 Oct 06 2021 13:06:09

%S 1,4620,89237148,2005604901300,47913489552349980,

%T 1183237138556438547120,29836408028165719837829700,

%U 763223193205837155576920270520,19728995249931089572476730815356700,514073874001824145407534840409364592528,13479596359042448208364688886016106250225648

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

%C From _Peter Bala_, Jan 24 2020: (Start)

%C 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).

%C 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)

%C 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

%H Gheorghe Coserea, <a href="/A295431/b295431.txt">Table of n, a(n) for n = 0..202</a>

%H F. Beukers and Heckman, G., <a href="http://eudml.org/doc/143655">Monodromy for the hypergeometric function nFn-1"</a>, Inventiones mathematicae 95.2 (1989): 325-354.

%H Jonathan Bober, <a href="http://arxiv.org/abs/0709.1977">Factorial ratios, hypergeometric series, and a family of step functions</a>, 2007, arXiv:0709.1977 [math.NT], 2007; J. London Math. Soc., Vol. 79, Issue 2 (2009), 422-444.

%H Gheorghe Coserea, <a href="/A295431/a295431.txt">Table with the parameters of the 52 sporadic integral factorial ratio sequences</a>

%H R. Mestrovic, <a href="http://arxiv.org/abs/1111.3057">Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011)</a>, arXiv:1111.3057 [math.NT], 2011.

%H F. Rodriguez-Villegas, <a href="http://arxiv.org/abs/math/0701362">Integral ratios of factorials and algebraic hypergeometric functions</a>, arXiv:math/0701362 [math.NT], 2007.

%H Wadim Zudilin, <a href="https://arxiv.org/abs/1901.07843">Congruences for q-binomial coefficients</a>, arXiv:1901.07843 [math.NT], 2019.

%F G.f.: hypergeom([1/12, 5/12, 7/12, 11/12], [1/3, 1/2, 2/3], 27648*x).

%F From _Karol A. Penson_, May 08 2018 (Start):

%F 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.

%F 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)

%F 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

%p seq((12*n)!*n!/((6*n)!*(4*n)!*(3*n)!),n=0..10); # _Karol A. Penson_, May 08 2018

%t Table[((12n)!n!)/((6n)!(4n)!(3n)!),{n,0,20}] (* _Harvey P. Dale_, Sep 14 2019 *)

%o (PARI)

%o r=[12,1]; s=[6,4,3];

%o p=[1/12,5/12,7/12,11/12]; q=[1/3,1/2,2/3];

%o C(r,s) = prod(k=1, #r, r[k]^r[k])/prod(k=1, #s, s[k]^s[k]);

%o u(r, s, N=20) = {

%o my(f=(v,n)->prod(k=1, #v, (v[k]*n)!));

%o apply(n->f(r,n)/f(s,n), [0..N-1]);

%o };

%o u(r,s,11)

%o \\ test 1:

%o \\ system("wget http://www.jjj.de/pari/hypergeom.gpi");

%o read("hypergeom.gpi");

%o N=200; x='x+O('x^N); u(r,s,N) == Vec(hypergeom(p, q, C(r,s)*x, N))

%o \\ test 2: check consistency of all parameters

%o system("wget https://oeis.org/A295431/a295431.txt");

%o N=200; x='x+O('x^N); w = read("a295431.txt");

%o 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))))

%Y The 52 sporadic integral factorial ratio sequences:

%Y Idx EntryID u(r,s) dFd-1

%Y ---+---------+--------------+-----------------------------------------------+

%Y 1 A295431 [12,1] [1/12,5/12,7/12,11/12]

%Y [6,4,3] [1/3,1/2,2/3]

%Y 2 A295432 [12,3,2] [1/12,5/12,7/12,11/12]

%Y [6,6,4,1] [1/6,1/2,5/6]

%Y 3 A295433 [12,1] [1/12,1/6,5/12,7/12,5/6,11/12]

%Y [8,3,2] [1/8,3/8,1/2,5/8,7/8]

%Y 4 A295434 [12,3] [1/12,1/3,5/12,7/12,2/3,11/12]

%Y [8,6,1] [1/8,3/8,1/2,5/8,7/8]

%Y 5 A295435 [12,3] [1/12,1/3,5/12,7/12,2/3,11/12]

%Y [6,5,4] [1/5,2/5,1/2,3/5,4/5]

%Y 6 A295436 [12,5] [1/12,1/6,5/12,7/12,5/6,11/12]

%Y [10,4,3] [1/10,3/10,1/2,7/10,9/10]

%Y 7 A295437 [18,1] [1/18,5/18,7/18,11/18,13/18,17/18]

%Y [9,6,4] [1/4,1/3,1/2,2/3,3/4]

%Y 8 A295438 [9,2] [1/9,2/9,4/9,5/9,7/9,8/9]

%Y [6,4,1] [1/6,1/4,1/2,3/4,5/6]

%Y 9 A295439 [9,4] [1/9,2/9,4/9,5/9,7/9,8/9]

%Y [8,3,2] [1/8,3/8,1/2,5/8,7/8]

%Y 10 A295440 [18,4,3] [1/18,5/18,7/18,11/18,13/18,17/18]

%Y [9,8,6,2] [1/8,3/8,1/2,5/8,7/8]

%Y 11 A295441 [9,1] [1/9,2/9,4/9,5/9,7/9,8/9]

%Y [5,3,2] [1/5,2/5,1/2,3/5,4/5]

%Y 12 A295442 [18,5,3] [1/18,5/18,7/18,11/18,13/18,17/18]

%Y [10,9,6,1] [1/10,3/10,1/2,7/10,9/10]

%Y 13 A295443 [18,4] [1/18,5/18,7/18,1/2,11/18,13/18,17/18]

%Y [12,9,1] [1/12,1/3,5/12,7/12,2/3,11/12]

%Y 14 A295444 [12,2] [1/12,1/6,5/12,1/2,7/12,5/6,11/12]

%Y [9,4,1] [1/9,2/9,4/9,5/9,7/9,8/9]

%Y 15 A295445 [18,2] [1/18,5/18,7/18,1/2,11/18,13/18,17/18]

%Y [9,6,5] [1/5,1/3,2/5,3/5,2/3,4/5]

%Y 16 A295446 [10,6] [1/10,1/6,3/10,1/2,7/10,5/6,9/10]

%Y [9,5,2] [1/9,2/9,4/9,5/9,7/9,8/9]

%Y 17 A295447 [14,3] [1/14,3/14,5/14,1/2,9/14,11/14,13/14]

%Y [9,7,1] [1/9,2/9,4/9,5/9,7/9,8/9]

%Y 18 A295448 [18,3,2] [1/18,5/18,7/18,1/2,11/18,13/18,17/18]

%Y [9,7,6,1] [1/7,2/7,3/7,4/7,5/7,6/7]

%Y 19 A295449 [12,2] [1/12,1/6,5/12,1/2,7/12,5/6,11/12]

%Y [7,4,3] [1/7,2/7,3/7,4/7,5/7,6/7]

%Y 20 A295450 [14,6,4] [1/14,3/14,5/14,1/2,9/14,11/14,13/14]

%Y [12,7,3,2] [1/12,1/3,5/12,7/12,2/3,11/12]

%Y 21 A295451 [14,1] [1/14,3/14,5/14,1/2,9/14,11/14,13/14]

%Y [7,5,3] [1/5,1/3,2/5,3/5,2/3,4/5]

%Y 22 A295452 [10,6,1] [1/10,1/6,3/10,1/2,7/10,5/6,9/10]

%Y [7,5,3,2] [1/7,2/7,3/7,4/7,5/7,6/7]

%Y 23 A295453 [15,1] [1/15,2/15,4/15,7/15,8/15,11/15,13/15,14/15]

%Y [9,5,2] [1/9,2/9,4/9,1/2,5/9,7/9,8/9]

%Y 24 A295454 [30,9,5] [1/30,7/30,11/30,13/30,17/30,19/30,23/30,29/30]

%Y [18,15,10,1] [1/18,5/18,7/18,1/2,11/18,13/18,17/18]

%Y 25 A295455 [15,4] [1/15,2/15,4/15,7/15,8/15,11/15,13/15,14/15]

%Y [12,5,2] [1/12,1/6,5/12,1/2,7/12,5/6,11/12]

%Y 26 A295456 [30,5,4] [1/30,7/30,11/30,13/30,17/30,19/30,23/30,29/30]

%Y [15,12,10,2] [1/12,1/3,5/12,1/2,7/12,2/3,11/12]

%Y 27 A295457 [15,4] [1/15,2/15,4/15,7/15,8/15,11/15,13/15,14/15]

%Y [8,6,5] [1/8,1/6,3/8,1/2,5/8,5/6,7/8]

%Y 28 A295458 [30,5,4] [1/30,7/30,11/30,13/30,17/30,19/30,23/30,29/30]

%Y [15,10,8,6] [1/8,1/3,3/8,1/2,5/8,2/3,7/8]

%Y 29 A295459 [15,2] [1/15,2/15,4/15,7/15,8/15,11/15,13/15,14/15]

%Y [10,4,3] [1/10,1/4,3/10,1/2,7/10,3/4,9/10]

%Y 30 A295460 [30,3,2] [1/30,7/30,11/30,13/30,17/30,19/30,23/30,29/30]

%Y [15,10,6,4] [1/5,1/4,2/5,1/2,3/5,3/4,4/5]

%Y 31 A211417 [30,1] [1/30,7/30,11/30,13/30,17/30,19/30,23/30,29/30]

%Y [15,10,6] [1/5,1/3,2/5,1/2,3/5,2/3,4/5]

%Y 32 A295462 [15,2] [1/15,2/15,4/15,7/15,8/15,11/15,13/15,14/15]

%Y [10,6,1] [1/10,1/6,3/10,1/2,7/10,5/6,9/10]

%Y 33 A295463 [15,7] [1/15,2/15,4/15,7/15,8/15,11/15,13/15,14/15]

%Y [14,5,3] [1/14,3/14,5/14,1/2,9/14,11/14,13/14]

%Y 34 A295464 [30,5,3] [1/30,7/30,11/30,13/30,17/30,19/30,23/30,29/30]

%Y [15,10,7,6] [1/7,2/7,3/7,1/2,4/7,5/7,6/7]

%Y 35 A295465 [30,5,3] [1/30,7/30,11/30,13/30,17/30,19/30,23/30,29/30]

%Y [15,12,10,1] [1/12,1/4,5/12,1/2,7/12,3/4,11/12]

%Y 36 A295466 [15,6,1] [1/15,2/15,4/15,7/15,8/15,11/15,13/15,14/15]

%Y [12,5,3,2] [1/12,1/4,5/12,1/2,7/12,3/4,11/12]

%Y 37 A295467 [15,1] [1/15,2/15,4/15,7/15,8/15,11/15,13/15,14/15]

%Y [8,5,3] [1/8,1/4,3/8,1/2,5/8,3/4,7/8]

%Y 38 A295468 [30,5,3,2] [1/30,7/30,11/30,13/30,17/30,19/30,23/30,29/30]

%Y [15,10,8,6,1] [1/8,1/4,3/8,1/2,5/8,3/4,7/8]

%Y 39 A295469 [20,3] [1/20,3/20,7/20,9/20,11/20,13/20,17/20,19/20]

%Y [12,10,1] [1/12,1/6,5/12,1/2,7/12,5/6,11/12]

%Y 40 A295470 [20,6,1] [1/20,3/20,7/20,9/20,11/20,13/20,17/20,19/20]

%Y [12,10,3,2] [1/12,1/3,5/12,1/2,7/12,2/3,11/12]

%Y 41 A295471 [20,1] [1/20,3/20,7/20,9/20,11/20,13/20,17/20,19/20]

%Y [10,8,3] [1/8,1/3,3/8,1/2,5/8,2/3,7/8]

%Y 42 A295472 [20,3,2] [1/20,3/20,7/20,9/20,11/20,13/20,17/20,19/20]

%Y [10,8,6,1] [1/8,1/6,3/8,1/2,5/8,5/6,7/8]

%Y 43 A061164 [20,1] [1/20,3/20,7/20,9/20,11/20,13/20,17/20,19/20]

%Y [10,7,4] [1/7,2/7,3/7,1/2,4/7,5/7,6/7]

%Y 44 A295474 [20,7,2] [1/20,3/20,7/20,9/20,11/20,13/20,17/20,19/20]

%Y [14,10,4,1] [1/14,3/14,5/14,1/2,9/14,11/14,13/14]

%Y 45 A295475 [20,3] [1/20,3/20,7/20,9/20,11/20,13/20,17/20,19/20]

%Y [10,9,4] [1/9,2/9,4/9,1/2,5/9,7/9,8/9]

%Y 46 A295476 [20,9,6] [1/20,3/20,7/20,9/20,11/20,13/20,17/20,19/20]

%Y [18,10,4,3] [1/18,5/18,7/18,1/2,11/18,13/18,17/18]

%Y 47 A295477 [24,1] [1/24,5/24,7/24,11/24,13/24,17/24,19/24,23/24]

%Y [12,8,5] [1/5,1/4,2/5,1/2,3/5,3/4,4/5]

%Y 48 A295478 [24,5,2] [1/24,5/24,7/24,11/24,13/24,17/24,19/24,23/24]

%Y [12,10,8,1] [1/10,1/4,3/10,1/2,7/10,3/4,9/10]

%Y 49 A295479 [24,4,1] [1/24,5/24,7/24,11/24,13/24,17/24,19/24,23/24]

%Y [12,8,7,2] [1/7,2/7,3/7,1/2,4/7,5/7,6/7]

%Y 50 A295480 [24,7,4] [1/24,5/24,7/24,11/24,13/24,17/24,19/24,23/24]

%Y [14,12,8,1] [1/14,3/14,5/14,1/2,9/14,11/14,13/14]

%Y 51 A295481 [24,4,3] [1/24,5/24,7/24,11/24,13/24,17/24,19/24,23/24]

%Y [12,9,8,2] [1/9,2/9,4/9,1/2,5/9,7/9,8/9]

%Y 52 A295482 [24,9,6,4] [1/24,5/24,7/24,11/24,13/24,17/24,19/24,23/24]

%Y [18,12,8,3,2] [1/18,5/18,7/18,1/2,11/18,13/18,17/18]

%Y Cf. A304126.

%K nonn

%O 0,2

%A _Gheorghe Coserea_, Nov 22 2017