I've been perusing the OEIS files and found that the series 1, 2, 1, -1, -2, 1, ... (my series divided by 3) can be generated by
evaluated at x = 0. (aka the coefficient of the nth term of the Taylor expansion of (1 + x)(1 - x + x^2) about x = 0.)
I can't seem to develop a pattern for the derivatives. Can anyone think of a way to use this information such that a pre-Calc student might be able to understand it?