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URL: https://oeis.org/A331545

⇱ A331545 - OEIS

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A331545
Triangle of constant term of the symmetric q-binomial coefficients.
0
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).
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
KEYWORD
nonn,tabl
AUTHOR
Michael Somos, Jan 19 2020
STATUS
approved