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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).

%I A128082 #19 Mar 17 2024 08:07:58
%S A128082 1,1,3,9,31,110,400,1477,5516,20775,78762,300179,1148995,4413877,
%T A128082 17007798,65707390,254430080,987162527,3836843836,14936223511,
%U A128082 58226118626,227271470103,888117198666,3474154716353,13603246639501,53310945927025,209093495360796
%N 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).
%H A128082 Alois P. Heinz, <a href="/A128082/b128082.txt">Table of n, a(n) for n = 0..500</a>
%F A128082 a(n+1) = A181971(2*n,n). - _Reinhard Zumkeller_, Jul 09 2012
%F A128082 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
%e A128082 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:
%e A128082    1;
%e A128082   (1);
%e A128082    1,(1),1;
%e A128082    1,2,(3),3,3,2,1;
%e A128082    1,3,6,(9),12,14,15,14,12,9,6,3,1;
%e A128082    1,4,10,19,(31),45,60,74,86,94,97,94,86,74,60,45,31,19,10,4,1;
%e A128082 The terms enclosed in parenthesis are initial terms of this sequence.
%p A128082 b:= proc(n) option remember; `if`(n=0, 1,
%p A128082       simplify(b(n-1)*(1-q^(2*n-1))/(1-q)))
%p A128082     end:
%p A128082 a:= n-> coeff(b(n+1), q, n):
%p A128082 seq(a(n), n=0..28);  # _Alois P. Heinz_, Sep 22 2021
%t A128082 a[n_] := SeriesCoefficient[Product[(1-q^(2k-1))/(1-q), {k, 1, n+1}], {q, 0, n}];
%t A128082 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Dec 31 2021 *)
%o A128082 (PARI) a(n)=if(n<1,0,polcoeff(prod(k=1,n,(1-q^(2*k-1))/(1-q)),n-1,q))
%Y A128082 Cf. A001147 ((2n-1)!!); A128080 (triangle), A128081 (central terms).
%K A128082 nonn
%O A128082 0,3
%A A128082 _Paul D. Hanna_, Feb 14 2007