asking for help

[Emeric Deutsch]

Dear seqfans,
The following is well-known:

If C(z)=[1-sqrt(1-4z)]/(2z) is the Catalan function, then
the coefficient of z^n in the expansion of C^q/sqrt(1-4z)
is binom(2n+q,n).

Is anything known about

the coefficient of z^n in the expansion of C^q/(1-4z)^(3/2) ?

For the cases q=1, q=2, q=3 there are formulas in
A000531, A029760, A045720, respectively.

Many thanks for any information you may have.

Emeric