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%I #5 Jan 19 2020 18:53:38
%S 1,1,1,1,0,1,1,1,1,1,1,0,2,0,1,1,1,2,2,1,1,1,0,3,0,3,0,1,1,1,3,5,5,3,
%T 1,1,1,0,4,0,8,0,4,0,1,1,1,4,8,12,12,8,4,1,1,1,0,5,0,18,0,18,0,5,0,1,
%U 1,1,5,13,24,32,32,24,13,5,1,1,1,0,6,0,33
%N Triangle of constant term of the symmetric q-binomial coefficients.
%C Symmetric q-binomial coefficients are based on symmetric q-numbers [n] := (q^n-1/q^n)/(q-1/q).
%F T(2*n, 2*k+1) = 0. T(2*n+1, 3) = A000982(n). T(2*n+1, 5) = A001973(n) if n>=2. T(4*n, 2*n) = A063074(n).
%e Triangle begins:
%e n\k| 0 1 2 3 4 5 6 7 ...
%e ---+----------------
%e 0 | 1
%e 1 | 1 1
%e 2 | 1 0 1
%e 3 | 1 1 1 1
%e 4 | 1 0 2 0 1
%e 5 | 1 1 2 2 1 1
%e 6 | 1 0 3 0 3 0 1
%e 7 | 1 1 3 5 5 3 1 1
%e ...
%t T[ n_, k_] := Coefficient[ QBinomial[ n, k, x^2] / x^(k (n - k)) // FunctionExpand // Expand, x, 0];
%o (PARI) {T(n, k) = if( k<0 || k>n, 0, polcoeff( prod(j = 1, k, (x^(n+1-j) - x^(-n-1+j))/(x^j - x^(-j))), 0))};
%Y CF. A000982, A001973, A063074, A188181.
%K nonn
%O 0,13
%A _Michael Somos_, Jan 19 2020
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