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A128085 Central coefficients of q in the q-analog of the even double factorials: a(n) = [q^([n^2/2])] Product_{j=1..n} (1-q^(2j))/(1-q).

%I A128085 #9 Feb 16 2025 08:33:04
%S A128085 1,1,2,8,46,340,3210,36336,484636,7394458,127707302,2454109404,
%T A128085 52091631896,1207854671388,30431260261770,826657521349952,
%U A128085 24114046688034516,751085176539860458,24899882719111953556
%N A128085 Central coefficients of q in the q-analog of the even double factorials: a(n) = [q^([n^2/2])] Product_{j=1..n} (1-q^(2j))/(1-q).
%C A128085 See A128081 for central coefficients of q in the q-analog of the odd double factorials. Also, A000140 is the central coefficients of q-factorials, giving the maximum number of permutations on n letters having the same number of inversions.
%H A128085 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/q-Factorial.html">q-Factorial</a>.
%e A128085 a(n) is the central term of the q-analog of even double factorials,
%e A128085 in which the coefficients of q (triangle A128084) begin:
%e A128085 n=0: (1);
%e A128085 n=1: (1),1;
%e A128085 n=2: 1,2,(2),2,1;
%e A128085 n=3: 1,3,5,7,(8),8,7,5,3,1;
%e A128085 n=4: 1,4,9,16,24,32,39,44,(46),44,39,32,24,16,9,4,1;
%e A128085 n=5: 1,5,14,30,54,86,125,169,215,259,297,325,(340),340,325,297,...;...
%e A128085 The terms enclosed in parenthesis are initial terms of this sequence.
%o A128085 (PARI) a(n)=if(n==0,1,polcoeff(prod(k=1,n,(1-q^(2*k))/(1-q)),n^2\2,q))
%Y A128085 Cf. A000165 ((2n)!!); A128084 (triangle), A128086 (diagonal); A128081.
%K A128085 nonn
%O A128085 0,3
%A A128085 _Paul D. Hanna_, Feb 14 2007