A128593 Column 1 of triangle A128592; a(n) = coefficient of q^(n+2) in the q-analog of the odd double factorials (2n+3)!! for n>=0.

1, 3, 12, 45, 170, 644, 2451, 9365, 35908, 138104, 532589, 2058782, 7975216, 30951921, 120326060, 468473348, 1826415556, 7129330988, 27860219331, 108984557708, 426730087879, 1672310507262, 6558840830680, 25742937514814, 101108341344396, 397368218111003

Offset: 0

Author

Paul D. Hanna, Mar 12 2007

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A128592; A128080; A001147 ((2n-1)!!); A128594 (column 2), A128595 (row sums).

Programs

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+2), q, n+2):
seq(a(n), n=0..30);   # Alois P. Heinz, Sep 22 2021

Mathematica

a[n_] := SeriesCoefficient[Product[(1-q^(2j-1))/(1-q), {j, 1, n+2}], {q, 0, n+2}];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jan 27 2024 *)

PARI

{a(n)=polcoeff(prod(j=1,n+2,(1-q^(2*j-1))/(1-q)),n+2,q)}

Formula

a(n) = [q^(n+2)] Product_{j=1..n+2} (1-q^(2j-1))/(1-q) for n>=0.