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

Question

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 $\sum_{i=0}^n a^n x^n$

Answer 1

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 $$ \sum_{k=0}^n \left( m+k\atop k \right) = \left( m+n+1 \atop n\right) $$