Pattern recognition ...

How to get to the solution?

One of the first things I did was to look at the ratios of successive terms. Letting $((a_n))$ denote the sequence, I saw

$\frac{a_2}{a_1} \approx 3$ ,
$\frac{a_3}{a_2} \approx 4$ , ...
$\frac{a_6}{a_5} \approx 7$ .

So this thing is growing super-exponentially. But the ratio of successive terms is growing almost linearly.

Next idea is to consider the differences $(a_{n} - (n+1)a_{n-1})$ to see how close this idea is:

$a_2 - 3a_1 = 4$ ,
$a_3 - 4a_2 = 5$ ,
$a_4 - 5a_2 = 6$ , etc.

This suggests the recursive formula $a_n = (n+1)a_{n-1} + (n+2)$ , initialized with $a_1 = 15$ . This relation works for all six numbers given.

In particular, if we extend this to $n=7$ :

$a_7 = 8 a_6 + 9 = 8*42519 + 9 = 340161$

The pattern of the form $ax + a + 1$ where $a$ increments by one each step. Here starts with $a=3$ . In the end:

$\boxed {42519 \times 8 + 9 = 340161}$