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A295431 a(n) = (12*n)!*n! / ((6*n)!*(4*n)!*(3*n)!). 52
1, 4620, 89237148, 2005604901300, 47913489552349980, 1183237138556438547120, 29836408028165719837829700, 763223193205837155576920270520, 19728995249931089572476730815356700, 514073874001824145407534840409364592528, 13479596359042448208364688886016106250225648 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

From Peter Bala, Jan 24 2020: (Start)

Supercongruences: 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 Mestrovoc, equation 39, p. 12).

More generally, for this sequence and the other integer factorial ratio sequences listed in the cross references, the supercongruences 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)

Supercongruences a(n*p) == a(n) ( mod p^3 ) are proven 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.

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.

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.

Sequence in context: A260054 A253115 A189983 * A338337 A237634 A051649

Adjacent sequences:  A295428 A295429 A295430 * A295432 A295433 A295434

KEYWORD

nonn

AUTHOR

Gheorghe Coserea, Nov 22 2017

STATUS

approved

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Last modified September 18 01:03 EDT 2021. Contains 347498 sequences. (Running on oeis4.)