Determine sequence with generating function $(2x-3)^3$

Question

Determine the sequence associated to the following generating function: $(2x-3)^3$

I know that the sequence $\left(\begin{pmatrix} 3 \\ 0 \end{pmatrix}, \begin{pmatrix} 3 \\ 1 \end{pmatrix}, \begin{pmatrix} 3 \\ 2 \end{pmatrix}, \ldots\right)$ has the generating function $(1+x)^3$ so I know that this sequence must be similar to the one I am looking for. But I am not sure how to manage the rest. Could someone help me?

Answer 1

The sequence determined by a power series $\sum_{n\ge 0}a_nx^n$ is simply the sequence $\langle a_0,a_1,a_2,\ldots\rangle$. Expanding $(2x-3)^3$, we get

$$-27+54x-36x^2+8x^3\;,$$

which corresponds to the sequence $\langle -27,54,-36,8,0,0,0,\ldots\rangle$.

Answer 2

The "generating function" for the series "$a_0, a_1, a_2, a_3,\cdot\cdot\cdot$" is the function, f(x), that corresponds to the power series $a_0+ a_1x+ a_2x^2+ a_3x^3+ \cdot\cdot\cdot$. In this case the function is a polynomial, $(2x- 3)^3= 8x^3- 36x^2+ 54x- 27$. Trivially then the sequence is -27, 54. -36, 8, 0, 0, 0, ...