# How can I determine the sequence generated by this generating function: $B(x)=(x+3)^2 + \frac{x}{(1-3x)^6}$?

How to find the sequence that is generated by this GF?

$$B(x)=(x+3)^2 + \frac{x}{(1-3x)^6}$$

We know that $$\frac{1}{(1-ax)}$$ is generated by $$\sum_{i=0}^n a^n x^n$$

You should remember that $$\frac{1}{(1-z)^m} = \sum_{n=0}^\infty \left( m+n-1\atop n\right) z^n$$ This can be proven by using induction and the fact that $$\sum_{k=0}^n \left( m+k\atop k \right) = \left( m+n+1 \atop n\right)$$
• that's $1/(1+z)^m$ ! – G Cab Feb 11 at 9:47