Suppose the EGF of $\{c_n\}_{n \geq 0}$ is $(e^x - 1)^3$. Find a formula for $c_n$.
2026-04-23 01:07:21.1776906441
Solving for Generating Functions
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Write out
$$(e^x-1)^3 = e^{3 x} - 3 e^{2 x} + 3 e^x -1$$
which has a series expansion
$$\sum_{n=0}^{\infty} \frac{x^n}{n!} [3^n - 3\cdot 2^n + 3 - \delta_{0n}]$$
Then
$$c_n = 3^n - 3\cdot 2^n + 3 - \delta_{0n}$$