A128082 A diagonal of the triangle A128080 of coefficients of q in the q-analog of the odd double factorials: a(n) = A128080(n+1,n).

1, 1, 3, 9, 31, 110, 400, 1477, 5516, 20775, 78762, 300179, 1148995, 4413877, 17007798, 65707390, 254430080, 987162527, 3836843836, 14936223511, 58226118626, 227271470103, 888117198666, 3474154716353, 13603246639501, 53310945927025, 209093495360796

Offset: 0

Author

Paul D. Hanna, Feb 14 2007

Keywords

• nonn

Examples

			a(n) is the n-th term in the q-analog of odd double factorial (2n+1)!!, in which the coefficients of q (triangle A128080) begin:
   1;
  (1);
   1,(1),1;
   1,2,(3),3,3,2,1;
   1,3,6,(9),12,14,15,14,12,9,6,3,1;
   1,4,10,19,(31),45,60,74,86,94,97,94,86,74,60,45,31,19,10,4,1;
The terms enclosed in parenthesis are initial terms of this sequence.
		

Links

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

Crossrefs

Cf. A001147 ((2n-1)!!); A128080 (triangle), A128081 (central terms).

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+1), q, n):
seq(a(n), n=0..28);  # Alois P. Heinz, Sep 22 2021

Mathematica

a[n_] := SeriesCoefficient[Product[(1-q^(2k-1))/(1-q), {k, 1, n+1}], {q, 0, n}];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Dec 31 2021 *)

PARI

a(n)=if(n<1,0,polcoeff(prod(k=1,n,(1-q^(2*k-1))/(1-q)),n-1,q))

Formula

a(n+1) = A181971(2*n,n). - Reinhard Zumkeller, Jul 09 2012

a(n) ~ c * 2^(2*n) / sqrt(n), where c = QPochhammer(1/2, 1/4) / sqrt(Pi) = 0.236633772766964806372497000634617466975260409008748... - Vaclav Kotesovec, Feb 07 2023, updated Mar 17 2024