Trinomial coefficients

The coefficient of xⁿ in the expansion of (1 + x + x²)^t is

Sum[Binomial[n + t - 1 - 3k, n - 3k] Binomial[t, k] (-1)^k, {k, 0, t}]

which can be evaluated as

Binomial[2t, n] Hypergeometric2F1[-n, n - 2t, 1/2 - t, 1/4]

or finally GegenbauerC[n, -t, -1/2]. This result follows directly from the generating function formula

(1 - 2 x z + x²)^-mSum[GegenbauerC[n, m, z] xⁿ, {n, 0, ∞}]