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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.

%I A128593 #12 Jan 27 2024 12:32:56
%S A128593 1,3,12,45,170,644,2451,9365,35908,138104,532589,2058782,7975216,
%T A128593 30951921,120326060,468473348,1826415556,7129330988,27860219331,
%U A128593 108984557708,426730087879,1672310507262,6558840830680,25742937514814,101108341344396,397368218111003
%N 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.
%H A128593 Alois P. Heinz, <a href="/A128593/b128593.txt">Table of n, a(n) for n = 0..500</a>
%F A128593 a(n) = [q^(n+2)] Product_{j=1..n+2} (1-q^(2j-1))/(1-q) for n>=0.
%p A128593 b:= proc(n) option remember; `if`(n=0, 1,
%p A128593       simplify(b(n-1)*(1-q^(2*n-1))/(1-q)))
%p A128593     end:
%p A128593 a:= n-> coeff(b(n+2), q, n+2):
%p A128593 seq(a(n), n=0..30);   # _Alois P. Heinz_, Sep 22 2021
%t A128593 a[n_] := SeriesCoefficient[Product[(1-q^(2j-1))/(1-q), {j, 1, n+2}], {q, 0, n+2}];
%t A128593 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Jan 27 2024 *)
%o A128593 (PARI) {a(n)=polcoeff(prod(j=1,n+2,(1-q^(2*j-1))/(1-q)),n+2,q)}
%Y A128593 Cf. A128592; A128080; A001147 ((2n-1)!!); A128594 (column 2), A128595 (row sums).
%K A128593 nonn
%O A128593 0,2
%A A128593 _Paul D. Hanna_, Mar 12 2007