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

0,13

Comments

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

Links

Table of n, a(n) for n=0..82.

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).

Example

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
   ...

Mathematica

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

Prog

(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))};

Crossrefs

CF. A000982, A001973, A063074, A188181.

Sequence in context: A158020 A051160 A051159 * A349465 A035697 A135549

Adjacent sequences: A331542 A331543 A331544 * A331546 A331547 A331548

Keyword

nonn

Author

Michael Somos, Jan 19 2020

Status

approved