Find sequence for generating function $\frac{1}{(1-x)^{12}}$

Question

I know that I should use partial fraction but is this really right approach, would not it be like 12 fractions? That power to 12 is something that is problematic for me. Can you give me some hint please?

EDIT: I forget to mention that formula for sequence should be withou infinity and sums, for examle: $a_n = 2n$ (and the sequence is made of coefficients that are at $x^n$)

Answer 1

Consider that: $$\sum_{n\geq 0} z^n = \frac{1}{1-z}, $$ then differentiate both terms eleven times. You will get: $$\sum_{n\geq 11} n(n-1)\cdot\ldots\cdot(n-10)\,z^{n-11}=\frac{11!}{(1-z)^{12}}$$ or: $$\frac{1}{(1-z)^{12}}=\sum_{n\geq 0}\binom{n+11}{11}z^n .$$