General form of a series

Question

I arrived at this series while solving some problem:

{-1, 14, -90, 350, -910, 1638, -2002, 1430, 0, -1430, 2002, -1638, 910, -350, 90, -14, 1}

Could anyone help me in knowing the general form/term of the n-th element in this series?

Or you could help me learn how to find the general term myself. I don't know where to start.

Thanks.

Answer 1

Starting with index $0$ this is $${(-1)}^{1+n}\left(\binom{14}{n}-\binom{14}{n-2}\right)$$ (using the fact that $\binom{N}m=0$ if $m<0$ or $m>N$.