<s>[INST] Create generating function: $(1, 1, -2, -2, 10, 3, -4, -4....)$ 
Find generating function (without using infinite series):
a) (0, 1, 4, 9, 16, 25, 36...)
b) (1, 1, -2, -2, 10, 3, -4, -4, 5, 5, -6, -6, 7...) (Only irregularity is the 10)

Here's what I got:
a) $(0, 1, 4, 9, 16...) = \frac{x(1+x)}{(1-x)^3}$
(Begin with (1, 1, 1, 1, ...) differentiate and shift by 1, then differentiate and shift by 1 again.)
Any help  with b)?
 [/INST] The function is
$$
7x^4+(1+x)(1-2x^2+3x^4-4x^6+\dots)
$$
Can you find a closed form for $1-2x^2+3x^4-4x^6+\dots$? Perhaps you can find a closed form for $1+2y+3y^2+4y^3+\dots$, then substitute $y\gets -x^2$?
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