A211421 Integral factorial ratio sequence: a(n) = (8*n)!*(3*n)!/((6*n)!*(4*n)!*n!).

1, 14, 390, 12236, 404550, 13777764, 478273692, 16825310040, 597752648262, 21397472070260, 770557136489140, 27884297395587240, 1013127645555452700, 36935287875280348776, 1350441573221798941560, 49498889739033621986736, 1818284097150186829038150

Offset: 0

Author

Peter Bala, Apr 10 2012

Keywords

• nonn
• easy

Comments

This sequence is the particular case a = 4, b = 3 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).

Sequence terms are given by the coefficient of x^n in the expansion of ( (1 + x)^(k+2)/(1 - x)^k )^n when k = 6. See the cross references for related sequences obtained from other values of k. - Peter Bala, Sep 29 2015

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..600
• Peter Bala, Notes on logarithmic differentiation, the binomial transform and series reversion
• J. W. Bober, Factorial ratios, hypergeometric series, and a family of step functions, 2007, arXiv:0709.1977v1 [math.NT], 2007; J. London Math. Soc., Vol. 79, Issue 2 (2009), 422-444.
• F. Rodriguez-Villegas, Integral ratios of factorials and algebraic hypergeometric functions, arXiv:math/0701362 [math.NT], 2007.

Crossrefs

Cf. A061162, A061163, A211419, A211420.

Cf. A000984 (k = 0), A091527 (k = 1), A001448 (k = 2), A262732 (k = 3), A211419 (k = 4), A262733 (k = 5), A262740.

Programs

Magma

[Factorial(8*n)*Factorial(3*n)/(Factorial(6*n)*Factorial(4*n)*Factorial(n)): n in [0..20]]; // Vincenzo Librandi, Aug 01 2016

Maple

#A211421
a := n -> (8*n)!*(3*n)!/((6*n)!*(4*n)!*n!);
seq(a(n), n = 0..16);

Mathematica

Table[(8 n)!*(3 n)!/((6 n)!*(4 n)!*n!), {n, 0, 15}] (* Michael De Vlieger, Oct 04 2015 *)

PARI

a(n) = (8*n)!*(3*n)!/((6*n)!*(4*n)!*n!);
vector(30, n, a(n-1)) \\ Altug Alkan, Oct 02 2015

Formula

The o.g.f. sum {n >= 1} a(n)*z^n is algebraic over the field of rational functions Q(z) (see Rodriguez-Villegas).

From Peter Bala, Sep 29 2015: (Start)

a(n) = Sum_{i = 0..n} binomial(8*n,i) * binomial(7*n-i-1,n-i).

a(n) = [x^n] ( (1 + x)^8/(1 - x)^6 )^n.

a(0) = 1 and a(n) = 2*(8*n - 1)*(8*n - 3)*(8*n - 5)*(8*n - 7)/( n*(6*n - 1)*(6*n - 3)*(6*n - 5) ) * a(n-1) for n >= 1.

exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 14*x + 293*x^2 + 7266*x^3 + 197962*x^4 + 5726364*x^5 + ... has integer coefficients and equals 1/x * series reversion of x*(1 - x)^6/(1 + x)^8. See A262740. (End)

a(n) ~ 2^(10*n)*27^(-n)/sqrt(2*Pi*n). - Ilya Gutkovskiy, Jul 31 2016

a(n) = (2^n/n!)*Product_{k = 3*n..4*n-1} (2*k + 1). - Peter Bala, Feb 26 2023

From Seiichi Manyama, Aug 09 2025: (Start)

a(n) = [x^n] 1/((1-x)^(n+1) * (1-2*x)^(6*n)).

a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(8*n,k) * binomial(2*n-k,n-k).

a(n) = Sum_{k=0..n} 2^k * binomial(6*n+k-1,k) * binomial(2*n-k,n-k).

a(n) = 4^n * binomial((8*n-1)/2,n).

a(n) = [x^n] 1/(1-4*x)^((6*n+1)/2).

a(n) = [x^n] (1+4*x)^((8*n-1)/2). (End)