cp's OEIS Frontend

A331545 Triangle of constant term of the symmetric q-binomial coefficients.

%I A331545 #5 Jan 19 2020 18:53:38
%S A331545 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 A331545 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 A331545 1,1,5,13,24,32,32,24,13,5,1,1,1,0,6,0,33
%N A331545 Triangle of constant term of the symmetric q-binomial coefficients.
%C A331545 Symmetric q-binomial coefficients are based on symmetric q-numbers [n] := (q^n-1/q^n)/(q-1/q).
%F A331545 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 A331545 Triangle begins:
%e A331545   n\k| 0 1 2 3 4 5 6 7  ...
%e A331545   ---+----------------
%e A331545    0 | 1
%e A331545    1 | 1 1
%e A331545    2 | 1 0 1
%e A331545    3 | 1 1 1 1
%e A331545    4 | 1 0 2 0 1
%e A331545    5 | 1 1 2 2 1 1
%e A331545    6 | 1 0 3 0 3 0 1
%e A331545    7 | 1 1 3 5 5 3 1 1
%e A331545    ...
%t A331545 T[ n_, k_] := Coefficient[ QBinomial[ n, k, x^2] / x^(k (n - k)) // FunctionExpand // Expand, x, 0];
%o A331545 (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 A331545 CF. A000982, A001973, A063074, A188181.
%K A331545 nonn
%O A331545 0,13
%A A331545 _Michael Somos_, Jan 19 2020