Create generating function: $(1, 1, -2, -2, 10, 3, -4, -4....)$

Question

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)?

Answer 1

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 $y\gets -x^2$?