Generating function of this sequence

Question

Find the generating function of the sequence with the property $\sum_{i=0}^n a_{i}a_{n-i} = 1$. I'm not sure where to start in this problem.

Answer 1

Hint: If $f(x) = \sum_{n=0}^\infty a_n x^n$ is a formal power series then $$ f(x)^2 = \sum_{n=0}^\infty \left(\sum_{i=0}^n a_{i}a_{n-i} \right) x^n $$ (See Cauchy product.) In your case, the right-hand side is a well-known infinite series.

Answer 2

Assume that: $$ f(x) = \sum_{n\geq 0}a_n x^n.\tag{1} $$ Cauchy's convolution hence gives: $$ f(x)^2 = \sum_{n\geq 0}\left(\sum_{i=0}^{n}a_i a_{n-i}\right) x^n = \sum_{n\geq 0} x^n = \frac{1}{1-x}\tag{2} $$ and we have $f(x)=\frac{1}{\sqrt{1-x}}$, from which: $$ a_n = \binom{-\frac{1}{2}}{n}(-1)^n=\frac{(2n-1)!!}{2^n\,n!}=\frac{1}{4^n}\binom{2n}{n}\approx\frac{1}{\sqrt{\pi n}}.\tag{3} $$