OFFSET
0,3
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..500
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
EXAMPLE
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.
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 *)
PROG
(PARI) a(n)=if(n<1, 0, polcoeff(prod(k=1, n, (1-q^(2*k-1))/(1-q)), n-1, q))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 14 2007
STATUS
approved