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

A368545 a(n) = (6*n)!*(9*n)!/((2*n)!*(3*n)!*((5*n)!)^2).

Original entry on oeis.org

1, 1512, 13477464, 156037527360, 2018989675062360, 27791880541579666512, 397974308267103144857280, 5857301482132559510090140920, 87960819866039245485380114158680, 1341476995616395775866469804150505600, 20709814202640945368698086671062023292464, 322897098547446926110838713561401344838974976
Offset: 0

Views

Author

Karol A. Penson, Dec 29 2023

Keywords

Crossrefs

Cf. A113424, A368513, A304126.

Programs

  • Maple
    seq((6*n)!*(9*n)!/((2*n)!*(3*n)!*((5*n)!)^2),n=0..12);

Formula

G.f.: hypergeometric10F9([1/9, 1/6, 2/9, 1/3, 4/9, 5/9, 2/3, 7/9, 5/6, 8/9], [1/5, 1/5, 2/5, 2/5, 3/5, 3/5, 4/5, 4/5, 1], (167365651248*z)/9765625).
E.g.f.: hypergeometric10F10([1/9, 1/6, 2/9, 1/3, 4/9, 5/9, 2/3, 7/9, 5/6, 8/9], [1/5, 1/5, 2/5, 2/5, 3/5, 3/5, 4/5, 4/5, 1,1], (167365651248*z)/9765625).
a(n) = Integral_{x=0..167365651248/9765625} x^n*W(x) dx, n>=0, where W(x) = (1953125*MeijerG([[], [-4/5, -4/5, -3/5, -3/5, -2/5, -2/5, -1/5, -1/5, 0, 0]], [[-1/9, -1/6, -2/9, -1/3, -4/9, -5/9, -2/3, -7/9, -5/6, -8/9], []], (9765625*x)/167365651248))/(111577100832*Pi). MeijerG is the Meijer G - function. This integral representation as the n-th power moment of the W(x) function in the interval [0, 167365651248/9765625] is verified by direct integration. However, we are unable at present to determine the geometric shape of W(x).

A368650 a(n) = (6*n + 1)!*(9*n + 1)!/((2*n)!*(3*n)!*((5*n + 1)!)^2).

Original entry on oeis.org

1, 2940, 27511848, 324265486545, 4234842288963000, 58626067532977225512, 842744763083824037236800, 12437726604034570811549435040, 187171833825593326056635733697560, 2859197188199406875783449346275416000, 44198453917285616202092687086145825181264, 689863061309915307698539343386922516078167200
Offset: 0

Views

Author

Karol A. Penson, Jan 02 2024

Keywords

Comments

a(n) can be rigorously proven to be an integer for n>=0.

Crossrefs

Cf. A304126, A368545, A082368, A113424.

Programs

  • Maple
    seq((6*n + 1)!*(9*n + 1)!/((2*n)!*(3*n)!*((5*n + 1)!)^2), n=0..12);

Formula

G.f.: hypergeometric10F9([2/9, 1/3, 4/9, 5/9, 2/3, 7/9, 5/6, 8/9, 10/9, 7/6], [2/5, 2/5, 3/5, 3/5, 4/5, 4/5, 1, 6/5, 6/5], (167365651248*z)/9765625).
O.g.f.: hypergeometric10F10([2/9, 1/3, 4/9, 5/9, 2/3, 7/9, 5/6, 8/9, 10/9, 7/6], [2/5, 2/5, 3/5, 3/5, 4/5, 4/5, 1, 1, 6/5, 6/5], (167365651248*z)/9765625).
a(n) = Integral_{x=0..167365651248/9765625} x^n*W(x) dx, n>=0, where W(x) = (78125*MeijerG([[], [-3/5, -3/5, -2/5, -2/5, -1/5, -1/5, 0, 0, 1/5, 1/5]], [[1/6, 1/9, -1/9, -1/6, -2/9, -1/3, -4/9, -5/9, -2/3, -7/9], []], (9765625*x)/167365651248))/(2066242608*Pi). MeijerG is the Meijer G - function. W(x) can be represented as a sum of 10 hypergeometric functions of type 10F9. W(x) can be proven to be a positive function in the interval [0, 167365651248/9765625]. W(x) is singular at x=0 and monotonically decreases to zero at x = 167365651248/9765625. This integral representation as the n-th power moment of the positive function W(x) in the interval [0, 167365651248/9765625] is unique, as W(x) is the solution of the Hausdorff moment problem.

A368692 a(n) = (12*n + 6)!*(6*n + 9)!/(108*(4*n + 2)!*(2*n + 3)!*((6*n + 5)!)^2).

Original entry on oeis.org

14, 563108, 54231252075, 6700034035890000, 928978310614152999200, 137569863175651804211692560, 21253098849879053645154605945160, 3381375421559384124434964404229384000, 549714622911935710495977183989400234273000
Offset: 0

Views

Author

Karol A. Penson, Jan 03 2024

Keywords

Comments

According to A. Adolphson and S. Sperber, "On the integrality of hypergeometric series whose coefficients are factorial ratios", ArXiv: 2001.03296, s.page 14, first equation after Eq.(7.4): for any two integers K, L, the ratios (3*K)!*(3*L)!/(K!*L!*((K+L)!)^2) are proven to be integers. 108*a(n) results from K = 4*n+2 and L = 2*n+3, n>=0. It is conjectured here that a(n) are integers.

Links

Crossrefs

Cf. A368650, A304126, A368545, A082368, A113424, A368875.

Programs

  • Maple
    seq((12*n + 6)!*(6*n + 9)!/(108*(4*n + 2)!*(2*n + 3)!*((6*n + 5)!)^2),n=0..9);

Formula

G.f.: 14*hypergeometric8F7([7/12, 2/3, 5/6, 11/12, 13/12, 17/12, 13/6, 7/3], [1, 7/6, 4/3, 3/2, 3/2, 5/3, 11/6], 186624*z).
E.g.f.: 14*hypergeometric8F8([7/12, 2/3, 5/6, 11/12, 13/12, 17/12, 13/6, 7/3], [1, 1, 7/6, 4/3, 3/2, 3/2, 5/3, 11/6], 186624*z).
a(n) = Integral_{x=0..186624} x^n*W(x) dx, n>=0, where W(x) = (1/(20736*Pi))*MeijerG([[], [0, 0, 1/6, 1/3, 1/2, 1/2, 2/3, 5/6]], [[-5/12, -1/3, -1/6, -1/12, 1/12, 5/12, 7/6, 4/3], []], x/186624). MeijerG is the Meijer G - function. W(x) can be represented as an expression containing the sum of 4 generalized hypergeometric functions of type 8F7. W(x) is a positive function in the interval [0, 186624], is singular at x=0 and monotonically decreases to zero at x = 186624. This integral representation as the n-th power moment of the positive function W(x) in the interval [0, 186624] is unique, as W(x) is the solution of the Hausdorff moment problem.
Let b(n) = Gamma(7+ 12*n)/(6*Gamma(2 + 2*n)*Gamma(3 + 4*n)*Gamma(6 + 6*n)), then a(n) = b(n) * A272399(n+2). - Peter Luschny, Jan 06 2024

A368875 a(n) = 24*(3*n + 1)!/(n!*((n + 2)!)^2).

Original entry on oeis.org

6, 16, 105, 1008, 12012, 164736, 2494206, 40646320, 701149020, 12655450080, 237026033790, 4577828250240, 90739095674400, 1838979005667840, 37993593597567210, 798259862714284080, 17022152442879594780, 367791659430639444000, 8040845154302354844450
Offset: 0

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Author

Karol A. Penson, Jan 08 2024

Keywords

Comments

According to A. Adolphson and S. Sperber (see Links), see page 14, second equation after Eq.(7.4): for any two integers K, L, the ratios (3*K+1)!*(3*L+1)!/(K!*L!*((K+L+1)!)^2) are proven to be integers. Here a(n) results from K = 1 and L = n, n >= 0.

Links

Crossrefs

Cf. A368650, A304126, A368545, A082368, A113424, A368692.

Programs

  • Maple
    seq(24*(3*n + 1)!/(n!*((n + 2)!)^2),n=0..17);
  • Mathematica
    Table[24*(3*n + 1)!/(n!*((n + 2)!)^2),{n,0,16}] (* James C. McMahon, Jan 08 2024 *)
  • SageMath
    def a(n): return (24 * (n + 1) * (n + 2) * gamma(3*n + 2)) / gamma(n + 3)^3
print([a(n) for n in range(19)])  # Peter Luschny, Jan 09 2024

Formula

G.f.: 6*hypergeometric3F2([2/3, 1, 4/3], [3, 3], 27*z).
G.f.: -(hypergeometric2F1([-4/3, -2/3], [1], 27*z) - 1)/(3*z^2) + 8/z.
E.g.f.: 6*hypergeometric3F3([2/3, 1, 4/3], [3, 3, 1], 27*z).
a(n) = Integral_{x=0..27} x^n*W(x) dx, n >= 0, where
W(x) = (243*2^(2/3)*Gamma(5/6)*Gamma(2/3)*hypergeometric2F1([-4/3, -4/3], [1/3], x/27)) / (16*Pi^(5/2)*x^(1/3)) - (3*sqrt(3)*2^(1/3)*x^(1/3)* hypergeometric2F1([-2/3, -2/3], [5/3], x/27))/(2*sqrt(Pi)*Gamma(5/6)* Gamma(2/3)).
W(x) is a positive function in the interval [0, 27], is singular at x = 0 with the singularity x^(-1/3), and monotonically decreases to zero at x = 27, with W'(x) tending to zero at x = 27. This integral representation as the n-th power moment of the positive function W(x) in the interval [0, 27] is unique, as W(x) is the solution of the Hausdorff moment problem.
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