OFFSET
0,4
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..150
FORMULA
a(n) ~ 3 * 2^n * n^(n - 3/2) / (sqrt(Pi) * exp(n)). - Vaclav Kotesovec, Feb 07 2023
EXAMPLE
a(n) is the central term of the q-analog of odd double factorials, in which the coefficients of q (triangle A128080) begin:
n=0: (1);
n=1: (1);
n=2: 1,(1),1;
n=3: 1,2,3,(3),3,2,1;
n=4: 1,3,6,9,12,14,(15),14,12,9,6,3,1;
n=5: 1,4,10,19,31,45,60,74,86,94,(97),94,86,74,60,45,31,19,10,4,1;
n=6: 1,5,15,34,65,110,170,244,330,424,521,614,696,760,801,(815),...;
The terms enclosed in parenthesis are initial terms of this sequence.
MAPLE
b:= proc(n) option remember; `if`(n=0, 1,
simplify(b(n-1)*(1-q^(2*n-1))/(1-q)))
end:
a:= n-> coeff(b(n), q, n*(n-1)/2):
seq(a(n), n=0..23); # Alois P. Heinz, Sep 22 2021
MATHEMATICA
a[n_Integer] := a[n] = Coefficient[Expand@Cancel@FunctionExpand[-q QPochhammer[1/q, q^2, n + 1]/(1 - q)^(n + 1)], q, n (n - 1)/2];
Table[a[n], {n, 0, 21}] (* Vladimir Reshetnikov, Sep 22 2021 *)
PROG
(PARI) a(n)=if(n==0, 1, polcoeff(prod(k=1, n, (1-q^(2*k-1))/(1-q)), n*(n-1)/2, q))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 14 2007
STATUS
approved