1, 2, 4, 8, 16, ... 33? Challenge

Write a function/program that outputs either the n'th element, or the first n elements, in the well known number sequence:

         1, 2, 4, 8, 16 ...


Oh, wait... I forgot the first few numbers:

1, 1, 1, 1, 2, 4, 8, 16 ...


Heck, I'll add a few more for good measure:

1, 1, 1, 1, 2, 4, 8, 16, 33, 69, 146, 312, 673, 1463, 3202, 7050, 15605, 34705 ...




The numbers are Generalized Catalan numbers given by the (zero-indexed) formula: 

$$a(n+1)= a(n) + \sum_{k=2}^{n-1} a(k)\cdot a(n-1-k)$$

where 

$$a(0)= a(1)= a(2)= a(3) = 1$$

This is OEIS A004149.

You may choose if you want to have the sequence zero- or one-indexed. The sequence must of course be the same, so you must rewrite the formula if you have it one-indexed.
Python, 51 bytes



f=lambda n,k=2:n&lt;3or k&lt;n and f(k)*f(n-k-2)+f(n,k+1)


Try it online!

Simplifies the formula a bit:

$$a(n) =  \sum_{k=2}^{n-1} a(k)\cdot a(n-2-k)$$

$$ a(-1) = a(0)= a(1)= a(2) = 1$$