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

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, 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, 1, 1, 5, 13, 24, 32, 32, 24, 13, 5, 1, 1, 1, 0, 6, 0, 33

Offset: 0

Author

Michael Somos, Jan 19 2020

Keywords

• nonn

Comments

Symmetric q-binomial coefficients are based on symmetric q-numbers [n] := (q^n-1/q^n)/(q-1/q).

Examples

			Triangle begins:
  n\k| 0 1 2 3 4 5 6 7  ...
  ---+----------------
   0 | 1
   1 | 1 1
   2 | 1 0 1
   3 | 1 1 1 1
   4 | 1 0 2 0 1
   5 | 1 1 2 2 1 1
   6 | 1 0 3 0 3 0 1
   7 | 1 1 3 5 5 3 1 1
   ...
		

Crossrefs

CF. A000982, A001973, A063074, A188181.

Programs

Mathematica

T[ n_, k_] := Coefficient[ QBinomial[ n, k, x^2] / x^(k (n - k)) // FunctionExpand // Expand, x, 0];

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))};

Formula

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).