OFFSET
0,2
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
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)
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 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Michel Lagneau, Sep 16 2012
STATUS
approved