A216702 a(n) = Product_{k=1..n} (16 - 4/k).

1, 12, 168, 2464, 36960, 561792, 8614144, 132903936, 2060011008, 32044615680, 499896004608, 7816555708416, 122459372765184, 1921670157238272, 30197673899458560, 475110069351481344, 7482983592285831168, 117967035454858985472, 1861257670509997326336

Offset: 0

Author

Michel Lagneau, Sep 16 2012

Keywords

• nonn
• easy

Comments

This sequence is generalizable: Product_{k=1..n} (q^2 - q/k) = (q^n/n!) * Product_{k=0..n-1} (q*k + q-1) = expansion of (1- x*q^2)^((1-q)/q).

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..800

Crossrefs

Cf. A004988, A049382, A004994, A034385, A098430.

Programs

Maple

seq(product(16-4/k, k=1.. n), n=0..20);
seq((4^n/n!)*product(4*k+3, k=0.. n-1), n=0..20);

Mathematica

Table[Product[16-4/k,{k,n}],{n,0,20}] (* or *) CoefficientList[ Series[ 1/(1-16*x)^(3/4),{x,0,20}],x] (* Harvey P. Dale, Sep 19 2012 *)

Formula

G.f.: 1/(1-16*x)^(3/4). - Harvey P. Dale, Sep 19 2012

From Peter Bala, Sep 24 2023: (Start)

a(n) = 16^n * binomial(n - 1/4, n).

P-recursive: a(n) = 4*(4*n - 1)/n * a(n-1) with a(0) = 1. (End)

From Peter Bala, Mar 31 2024: (Start)

a(n) = (-16)^n * binomial(-3/4, n).

a(n) ~ 1/Gamma(3/4) * 16^n/n^(1/4).

E.g.f.: hypergeom([3/4], [1], 16*x).

a(n) = (16^n)*Sum_{k = 0..2*n} (-1)^k*binomial(-3/4, k)* binomial(-3/4, 2*n - k).

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

Sum_{k = 0..n} a(k)*a(n-k) = (16^n)/(2*n)! * Product_{k = 1..n} (4*k^2 - 1) = (16^n)/(2*n)! * A079484(n). (End)