Challenge Question: Number Sequence!?

[Guru of a Thousand Songs]

Just a sequences I thought up. 10 points for the first correct answer as to the next number (hopefully with explanation).

2, 24, 126, 416, 1050

[Helmut]

One possibility:
. . . 2
. . 24 . 22
. 126 102 . 80
. 416 290 188 108
1050 634 344 156 48
1684 978 500 204 48

Another way:
y(x) = 2x^2 - 2x^3 + 2x^4 =
y(x) = 2x^2(1 - x(1 - x))
y(6) = 72(1 - 6(1 - 6)) = 2232

[Dr D]

This is actually a perfect quartic fit.

f(n) = 2n^4 + 6n^3 + 8n^2 + 6n + 2
where n starts from 0

The next term should be 2232

This could be factorized to give:
f(n) = 2*[(1+n)^4 - n*(1+n)^2]
= 2*(1+n)^2 * (1+n+n^2)