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A128592 Triangle, read by rows, of coefficients of q^(nk+k) in the q-analog of the odd double factorials: T(n,k) = [q^(nk+k)] Product_{j=1..n+1} (1-q^(2j-1))/(1-q) for n>0, with T(0,0)=1.

%I A128592 #8 Sep 27 2022 14:36:18
%S A128592 1,1,1,1,3,1,1,12,12,1,1,45,97,45,1,1,170,696,696,170,1,1,644,4784,
%T A128592 8447,4784,644,1,1,2451,32230,92003,92003,32230,2451,1,1,9365,214978,
%U A128592 946330,1487477,946330,214978,9365,1,1,35908,1426566,9417798,21856230,21856230
%N A128592 Triangle, read by rows, of coefficients of q^(nk+k) in the q-analog of the odd double factorials: T(n,k) = [q^(nk+k)] Product_{j=1..n+1} (1-q^(2j-1))/(1-q) for n>0, with T(0,0)=1.
%F A128592 T(n,k) = A128080(n+1,nk+k) where A128080 is the triangle of coefficients of q in the q-analog of the odd double factorials.
%e A128592 Triangle begins:
%e A128592 1;
%e A128592 1, 1;
%e A128592 1, 3, 1;
%e A128592 1, 12, 12, 1;
%e A128592 1, 45, 97, 45, 1;
%e A128592 1, 170, 696, 696, 170, 1;
%e A128592 1, 644, 4784, 8447, 4784, 644, 1;
%e A128592 1, 2451, 32230, 92003, 92003, 32230, 2451, 1;
%e A128592 1, 9365, 214978, 946330, 1487477, 946330, 214978, 9365, 1;
%e A128592 1, 35908, 1426566, 9417798, 21856230, 21856230, 9417798, 1426566, 35908, 1;
%e A128592 1, 138104, 9441417, 91852376, 302951392, 441170745, 302951392, 91852376, 9441417, 138104, 1;
%t A128592 T[n_, k_] := If[k < 0 || k > n*(n + 1), 0, If[n == 0, 1, SeriesCoefficient[Product[(1 - q^(2*j - 1))/(1 - q), {j, 1, n + 1}], {q, 0, (n + 1)*k}]]];
%t A128592 Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Sep 27 2022, from PARI code *)
%o A128592 (PARI) T(n,k)=if(k<0 || k>n*(n+1),0,if(n==0,1, polcoeff(prod(j=1,n+1,(1-q^(2*j-1))/(1-q)),(n+1)*k,q)))
%Y A128592 Cf. A128080; A001147 ((2n-1)!!); A128593 (column 1), A128594 (column 2), A128595 (row sums); variant: A128596.
%K A128592 nonn,tabl
%O A128592 0,5
%A A128592 _Paul D. Hanna_, Mar 12 2007