A128081 Central coefficients of q in the q-analog of the odd double factorials: a(n) = [q^(n(n-1)/2)] Product_{j=1..n} (1-q^(2j-1))/(1-q).

1, 1, 1, 3, 15, 97, 815, 8447, 104099, 1487477, 24188525, 441170745, 8920418105, 198066401671, 4791181863221, 125421804399845, 3532750812110925, 106538929613501939, 3425126166609830467, 116938867144129019137, 4225543021235970185429, 161113285522023566327031

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

Cf. A001147 ((2n-1)!!); A128080 (triangle), A128082 (diagonal).

Sequence in context: A132437 A331610 A331618 * A370676 A186264 A140286

Adjacent sequences: A128078 A128079 A128080 * A128082 A128083 A128084

Keyword

nonn

Author

Paul D. Hanna, Feb 14 2007

Status

approved