cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Original entry on oeis.org

1, 4620, 89237148, 2005604901300, 47913489552349980, 1183237138556438547120, 29836408028165719837829700, 763223193205837155576920270520, 19728995249931089572476730815356700, 514073874001824145407534840409364592528, 13479596359042448208364688886016106250225648
Offset: 0

Views

Author

Gheorghe Coserea, Nov 22 2017

Keywords

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

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.

Programs

  • 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 *)
  • 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))))

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

A061162 a(n) = (6n)!n!/((3n)!(2n)!^2).

Original entry on oeis.org

1, 30, 2310, 204204, 19122246, 1848483780, 182327718300, 18236779032600, 1842826521244230, 187679234340049620, 19232182592635611060, 1980665038436368775400, 204826599735691440534300, 21255328931341321610645544, 2212241139727064219063537016
Offset: 0

Views

Author

Richard Stanley, Apr 17 2001

Keywords

Comments

According to page 781 of the cited reference the generating function F(x) for a(n) is algebraic but not obviously so and the minimal polynomial satisfied by F(x) is quite large.
This sequence is the particular case a = 3, b = 1 of the following result (see Bober, Theorem 1.2): let a, b be nonnegative integers with a > b and gcd(a,b) = 1. Then (2*a*n)!*(b*n)!/((a*n)!*(2*b*n)!*((a-b)*n)!) is an integer for all integer n >= 0. Other cases include A211419 (a = 3, b = 2), A211420 (a = 4, b = 1) and A211421 (a = 4, b = 3) and A061163 (a = 5, b = 1). The o.g.f. Sum_{n >= 1} a(n)*z^n is algebraic over the field of rational functions Q(z) (see Rodriguez-Villegas). - Peter Bala, Apr 10 2012

References

  • M. Kontsevich and D. Zagier, Periods, in Mathematics Unlimited - 2001 and Beyond, Springer, Berlin, 2001, pp. 771-808.
  • R. P. Stanley, Enumerative Combinatorics Volume 2, Cambridge Univ. Press, 1999, Theorem 6.33, p. 197.

Links

Crossrefs

Cf. A061163, A061164, A007297, A091527, A262732.
See also A091527, A211419, A211420, A211421.

Programs

  • Maple
    A061162 := n->(6*n)!*n!/((3*n)!*(2*n)!^2);
  • Mathematica
    a[n_] := 16^n Gamma[3 n + 1/2]/(Gamma[n + 1/2] Gamma[2 n + 1]);
Table[a[n], {n, 0, 14}] (* Peter Luschny, Mar 01 2018 *)
  • PARI
    { for (n=0, 100, write("b061162.txt", n, " ", (6*n)!*n!/((3*n)!*(2*n)!^2)) ) } \\ Harry J. Smith, Jul 18 2009

Formula

a(n) ~ 1/2*Pi^(-1/2)*n^(-1/2)*2^(2*n)*3^(3*n)*{1 - 1/72*n^-1 + ...}. - Joe Keane (jgk(AT)jgk.org), Jun 11 2002
n*(2*n-1)*a(n) -6*(6*n-1)*(6*n-5)*a(n-1)=0. - R. J. Mathar, Oct 26 2014
From Peter Bala, Aug 21 2016: (Start)
a(n) = Sum_{k = 0..2*n} binomial(6*n, k)*binomial(4*n - k - 1, 2*n - k).
a(n) = Sum_{k = 0..n} binomial(8*n, 2*n - 2*k)*binomial(2*n + k - 1, k).
O.g.f. A(x) = Hypergeom([5/6, 1/6], [1/2], 108*x).
a(n) = [x^(2*n)] H(x)^n, where H(x) = (1 + x)^6/(1 - x)^2. Cf. A091496 and A262732. It follows that the o.g.f. A(x) for this sequence is the diagonal of the bivariate rational generating function 1/2*( 1/(1 - t*H(sqrt(x))) + 1/(1 - t*H(-sqrt(x))) ) and hence is algebraic by Stanley 1999, Theorem 6.33, p. 197.
Let G(x) = 1/x * series reversion( x*(1 - x)/(1 + x)^3 ) = 1 + 4*x + 23*x^2 + 156*x^3 + 1162*x^4 + ..., essentially the o.g.f. for A007297. Then A(x^2) equals the even part of 1 + x*(d/dx log(G(x))).
exp(Sum_{n >= 1} a(n)*x^n/n) = F(x), where F(x) = 1 + 30*x + 1605*x^2 + 107218*x^3 + 8043114*x^4 + 647773116*x^5 + 54730094637*x^6 + ... has integer coefficients since F(x^2) = G(x)*G(-x). Furthermore, F(x)^(1/6) = 1 + 5*x + 205*x^2 + 12328*x^3 + 874444*x^4 + 68022261*x^5 + 5613007167*x^6 + ... appears to have all integer coefficients. (End)
a(n) is the n-th moment of the positive weight function w(x) on x = (0,108), i.e.: a(n) = Integral_{x=0..108} x^n*w(x) dx, n >= 0, where w(x) = sqrt(3)*(1 + sqrt(1 - x/108))^(2/3)/(12*2^(1/3)*Pi*x^(5/6)*sqrt(1 - x/108)) + 2^(4/3)*sqrt(3)/(864*Pi*x^(1/6)*(1 + sqrt(1 - x/108))^(2/3)*sqrt(1 - x/108)). The weight function w(x) is singular at x=0 and at x=108 and is the solution of the Hausdorff moment problem. This solution is unique. - Karol A. Penson, Mar 01 2018
a(n) = 2^(4*n)*binomial(-n-1/2, 2*n). - Ira M. Gessel, Jan 04 2025

A061163 a(n) = (10n)!*n!/((5n)!*(4n)!*(2n)!).

Original entry on oeis.org

1, 630, 1385670, 3528923580, 9540949030470, 26651569523959380, 75998432812419471900, 219813190240007470094520, 642409325786050322446410310, 1892390644737640220059489996260
Offset: 0

Views

Author

Richard Stanley, Apr 17 2001

Keywords

Comments

According to page 781 of the cited reference the generating function F(x) for a(n) is algebraic but not obviously so and the minimal polynomial satisfied by F(x) is quite large.
This sequence is the particular case a = 5, b = 1 of the following result (see Bober, Theorem 1.2): let a, b be nonnegative integers with a > b and GCD(a,b) = 1. Then (2*a*n)!*(b*n)!/((a*n)!*(2*b*n)!*((a-b)*n)!) is an integer for all integer n >= 0. Other cases include A061162 (a = 3, b = 1), A211419 (a = 3, b = 2) and A211420(a = 4, b = 1) and A211421 (a = 4, b = 3). The o.g.f. Sum_{n >= 1} a(n)*z^n is algebraic over the field of rational functions Q(z) (see Rodriguez-Villegas). - Peter Bala, Apr 10 2012
Continuing the comment above: This is case n = 4 of the array of sequences
A(n, k) = 4^(n*k)*(Gamma((n + 1)*k + 1/2)/Gamma(k + 1/2)) / Gamma(n * k + 1). See the cross-references for other cases. - Peter Luschny, Feb 21 2024

References

  • M. Kontsevich and D. Zagier, Periods, in Mathematics Unlimited - 2001 and Beyond, Springer, Berlin, 2001, pp. 771-808.

Links

Crossrefs

Cf. A061164, A211419, A211421.
Cf. A000012 (n=0), A001448 (n=1), A061162 (n=2), A211420 (n=3), this sequence (n=4).

Programs

  • Maple
    A061163 := n->(10*n)!*n!/((5*n)!*(4*n)!*(2*n)!);
# Alternative:
A := (n, k) -> 4^(n*k)*(GAMMA((n + 1)*k + 1/2)/GAMMA(k + 1/2))/GAMMA(n*k + 1):
seq(A(4, k), k = 0..9);  # Peter Luschny, Feb 21 2024
  • Mathematica
    Table[(10n)! n!/((5n)!(4n)!(2n)!),{n,0,10}] (* Harvey P. Dale, Oct 24 2022 *)

Formula

n*(4*n-3)*(2*n-1)*(4*n-1)*a(n) -10*(10*n-9)*(10*n-7)*(10*n-3)*(10*n-1)*a(n-1)=0. - R. J. Mathar, Oct 26 2014
O.g.f. is a generalized hypergeometric function 4F3([1/10, 3/10, 7/10, 9/10], [1/4, 1/2, 3/4], 5^5*z). - Karol A. Penson, Apr 13 2022
From Karol A. Penson, Feb 21 2024: (Start)
(O.g.f.(z))^2 satisfies the algebraic equation of order 15, in which the powers of (O.g.f.(z))^2 are multiplied by polynomials p(n, z) with integer coefficients, in the form: Sum_{n = 0..15} p(n, z)*(O.g.f.(z))^(2*n) = 0.
Here is the list of orders, in the variable z, of all polynomials p(n, z) for n=0..15: 9,9,9,9,9,10,10,10,10,10,10,11,11,11,11,11,12. For example p(15, z) = 2^50*(5^5*z-1)^12. (End)
a(n) ~ 5^(5*n) / (2^(3/2) * sqrt(Pi*n)). - Vaclav Kotesovec, Aug 27 2024

A211417 Integral factorial ratio sequence: a(n) = (30*n)!*n!/((15*n)!*(10*n)!*(6*n)!).

Original entry on oeis.org

1, 77636318760, 53837289804317953893960, 43880754270176401422739454033276880, 38113558705192522309151157825210540422513019720, 34255316578084325260482016910137568877961925210286281393760
Offset: 0

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Author

Peter Bala, Apr 11 2012

Keywords

Comments

The integrality of this sequence can be used to prove Chebyshev's estimate C(1)*x/log(x) <= #{primes <= x} <= C(2)*x/log(x), for x sufficiently large; the constant C(1) = 0.921292... and C(2) = 1.105550.... Chebyshev's approach used the related step function floor(x) -floor(x/2) -floor(x/3) -floor(x/5) +floor(x/30). See A182067.
This sequence is one of the 52 sporadic integral factorial ratio sequences of height 1 found by V. I. Vasyunin.
The o.g.f. sum {n >= 0} a(n)*z^n is a generalized hypergeometric series of type 8F7 (see Bober, Table 2, Entry 31) and is an algebraic function of degree 483840 over the field of rational functions Q(z) (see Rodriguez-Villegas). Bober remarks that the monodromy group of the differential equation satisfied by the o.g.f. is W(E_8), the Weyl group of the E_8 root system.
See the Bala link for the proof that a(n), n = 0,1,2..., is an integer.
Congruences: 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(30*n,15*n)*C(15*n,5*n)/C(6*n,n) and use equation 39 in Mestrovic, p. 12). More generally, 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. Cf. A295431. - Peter Bala, Jan 24 2020

Links

Crossrefs

Cf. A182067, A211418, A061162, A061163, A061164, A091496, A091527, A112292, A182400, A211419, A211420, A211421, A276100, A262733, A295431.

Programs

  • Magma
    [Factorial(30*n)*Factorial(n)/(Factorial(15*n)*Factorial(10*n)*Factorial(6*n)): n in [0..10]]; // Vincenzo Librandi, Oct 03 2015
  • Mathematica
    Table[(30 n)!*n!/((15 n)!*(10 n)!*(6 n)!), {n, 0, 5}] (* Michael De Vlieger, Oct 02 2015 *)
  • PARI
    a(n) = (30*n)!*n!/((15*n)!*(10*n)!*(6*n)!);
vector(10, n, a(n-1)) \\ Altug Alkan, Oct 02 2015

Formula

a(n) ~ 2^(14*n-1) * 3^(9*n-1/2) * 5^(5*n-1/2) / sqrt(Pi*n). - Vaclav Kotesovec, Aug 30 2016

A364180 a(n) = (10*n)!*(n/2)!/((5*n)!*(7*n/2)!*(2*n)!).

Original entry on oeis.org

1, 1152, 5542680, 31473008640, 190818980609400, 1198265754978353152, 7691041400616850556280, 50107639155283424528302080, 330014847932376708502470210680, 2191489080600524699617120065945600, 14647137653300940580784413641872332680
Offset: 0

Views

Author

Peter Bala, Jul 13 2023

Keywords

Comments

A061164, defined by A061164(n) = (20*n)!*n! / ((10*n)!*(7*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 43). Here we are essentially considering the sequence {A061164(n/2) : n >= 0}. Fractional factorials are defined in terms of the gamma function; for example, (7*n/2)! := Gamma(1 + 7*n/2).
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.

Links

Crossrefs

Cf. A061164, A276100, A276101, A276102, A295431, A347854, A347855, A347856, A347857, A347858, A364173 - A364185.

Programs

  • Maple
    seq( simplify((10*n)!*(n/2)!/((5*n)!*(7*n/2)!*(2*n)!)), n = 0..15);

Formula

a(n) ~ c^n * 1/sqrt(14*Pi*n), where c = (2^11)*(5^5)/(7^4) * sqrt(7).
a(n) = 409600*(10*n - 1)*(10*n - 3)*(10*n - 7)*(10*n - 9)*(10*n - 11)*(10*n - 13)*(10*n - 17)*(10*n - 19)/(7*n*(n - 1)*(7*n - 2)*(7*n - 4)*(7*n - 6)*(7*n - 8)*(7*n - 10)*(7*n - 12))*a(n-2) with a(0) = 1 and a(1) = 1152.
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