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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).

%I A128081 #22 Feb 07 2023 09:58:19
%S A128081 1,1,1,3,15,97,815,8447,104099,1487477,24188525,441170745,8920418105,
%T A128081 198066401671,4791181863221,125421804399845,3532750812110925,
%U A128081 106538929613501939,3425126166609830467,116938867144129019137,4225543021235970185429,161113285522023566327031
%N 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).
%H A128081 Alois P. Heinz, <a href="/A128081/b128081.txt">Table of n, a(n) for n = 0..150</a>
%F A128081 a(n) ~ 3 * 2^n * n^(n - 3/2) / (sqrt(Pi) * exp(n)). - _Vaclav Kotesovec_, Feb 07 2023
%e A128081 a(n) is the central term of the q-analog of odd double factorials, in which the coefficients of q (triangle A128080) begin:
%e A128081    n=0: (1);
%e A128081    n=1: (1);
%e A128081    n=2: 1,(1),1;
%e A128081    n=3: 1,2,3,(3),3,2,1;
%e A128081    n=4: 1,3,6,9,12,14,(15),14,12,9,6,3,1;
%e A128081    n=5: 1,4,10,19,31,45,60,74,86,94,(97),94,86,74,60,45,31,19,10,4,1;
%e A128081    n=6: 1,5,15,34,65,110,170,244,330,424,521,614,696,760,801,(815),...;
%e A128081 The terms enclosed in parenthesis are initial terms of this sequence.
%p A128081 b:= proc(n) option remember; `if`(n=0, 1,
%p A128081       simplify(b(n-1)*(1-q^(2*n-1))/(1-q)))
%p A128081     end:
%p A128081 a:= n-> coeff(b(n), q, n*(n-1)/2):
%p A128081 seq(a(n), n=0..23);  # _Alois P. Heinz_, Sep 22 2021
%t A128081 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];
%t A128081 Table[a[n], {n, 0, 21}] (* _Vladimir Reshetnikov_, Sep 22 2021 *)
%o A128081 (PARI) a(n)=if(n==0,1,polcoeff(prod(k=1,n,(1-q^(2*k-1))/(1-q)),n*(n-1)/2,q))
%Y A128081 Cf. A001147 ((2n-1)!!); A128080 (triangle), A128082 (diagonal).
%K A128081 nonn
%O A128081 0,4
%A A128081 _Paul D. Hanna_, Feb 14 2007