Is it possible for

centered trinomial coefficients to be computed in O(log n) time, analogous to how the time complexity of modular exponentiation is O(log n)?

I am well aware that factorials cannot be computed in logarithmic time, making

Wilson's theorem impractical for large numbers.

On the other hand, raising an integer to a power can easily be done in logarithmic time, making

Fermat's test practical for large numbers.

Where does computing centered trinomial coefficient (and similar sequences), fall into?

To be clear, central trinomial coefficients are the coefficient of x^n in (x^2 + x + 1)^n for n ≥ 0.

Meanwhile, centered binomial coefficients binomial(2*n, n) are the coefficient of x^n in (x^2 + 2x + 1)^n for n ≥ 0.

Catalan numbers have the form binomial(2*n, n)/(n + 1) for n ≥ 0. I can't name a single algorithm for computing either central binomial coefficients or Catalan numbers in O(log n) time.

Is the same true for centered trinomial coefficients, and if so, why is it impossible to compute any particular term in O(log n) time?