Find the $n^{th}$ term for this sequence

I have the following sequence:

4, 7, 11, 19, 36, 69

Now, I've done the usual and found the differences, and it goes down to four levels until I get a common difference, suggesting I have to use $n^{4}$ somewhere, but I just can't find the nth term. Any help? And if you know of any easier methods to finding the nth term, I'd appreciate it.

P.S. - I know about the formulas for arithmetic progression and geometric progression, but clearly neither can be used here.

Answer

The first numbers in each row of differences are 4, 3, 1, 3, 2. Assuming that the last row is all 2's then the formula is
$$4\binom n0+ 3\binom n1 + 1\binom n2 + 3\binom n3 + 2\binom n4$$

starting at $n=0$.
This is an instance of the Newton series.

The formula simplifies to
$$\frac{n^4 - n^2 + 36 n + 48}{12}$$
but this is not enlightening.