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A128596 Triangle, read by rows, of coefficients of q^(nk) in the q-analog of the even double factorials: T(n,k) = [q^(nk)] Product_{j=1..n} (1-q^(2j))/(1-q) for n>0, with T(0,0)=1.

%I A128596 #5 Jun 14 2017 00:24:52
%S A128596 1,1,1,1,2,1,1,7,7,1,1,24,46,24,1,1,86,297,297,86,1,1,315,1919,3210,
%T A128596 1919,315,1,1,1170,12399,32510,32510,12399,1170,1,1,4389,80241,318171,
%U A128596 484636,318171,80241,4389,1,1,16588,520399,3054100,6730832,6730832
%N A128596 Triangle, read by rows, of coefficients of q^(nk) in the q-analog of the even double factorials: T(n,k) = [q^(nk)] Product_{j=1..n} (1-q^(2j))/(1-q) for n>0, with T(0,0)=1.
%F A128596 T(n,k) = A128084(n,nk) where A128084 is the triangle of coefficients of q in the q-analog of the even double factorials.
%e A128596 Row sums equal 2*A000165(n-1) for n>0, twice the even double factorials:
%e A128596 [1, 2, 4, 16, 96, 768, 7680, 92160, 1290240, ..., 2*(2n-2)!!, ...].
%e A128596 Triangle begins:
%e A128596 1;
%e A128596 1, 1;
%e A128596 1, 2, 1;
%e A128596 1, 7, 7, 1;
%e A128596 1, 24, 46, 24, 1;
%e A128596 1, 86, 297, 297, 86, 1;
%e A128596 1, 315, 1919, 3210, 1919, 315, 1;
%e A128596 1, 1170, 12399, 32510, 32510, 12399, 1170, 1;
%e A128596 1, 4389, 80241, 318171, 484636, 318171, 80241, 4389, 1;
%e A128596 1, 16588, 520399, 3054100, 6730832, 6730832, 3054100, 520399, 16588, 1;
%e A128596 1, 63064, 3382588, 28980565, 89514691, 127707302, 89514691, 28980565, 3382588, 63064, 1;
%o A128596 (PARI) T(n,k)=if(k<0 || k>n^2,0,if(n==0,1,polcoeff(prod(j=1,n,(1-q^(2*j))/(1-q)),n*k,q)))
%Y A128596 Cf. A128084; A000165 ((2n)!!); A128086 (column 1), A128597 (column 2), A128598 (column 3); variant: A128592.
%K A128596 nonn,tabl
%O A128596 0,5
%A A128596 _Paul D. Hanna_, Mar 12 2007