grandes-ecoles 2021 Q12

grandes-ecoles · France · centrale-maths1__mp Sequences and Series Functional Equations and Identities via Series

For every integer $n \geqslant 1$, we denote $C_{n}$ the number of well-parenthesized words of length $2n$. We set by convention $C_{0} = 1$. For every $x \in ]-\frac{1}{4}, \frac{1}{4}[$, we set $F(x) = \sum_{k=0}^{+\infty} C_{k} x^{k}$.
Show that, for every $x \in ]-\frac{1}{4}, \frac{1}{4}[$, $F(x) = 1 + x(F(x))^{2}$.

For every integer $n \geqslant 1$, we denote $C_{n}$ the number of well-parenthesized words of length $2n$. We set by convention $C_{0} = 1$. For every $x \in ]-\frac{1}{4}, \frac{1}{4}[$, we set $F(x) = \sum_{k=0}^{+\infty} C_{k} x^{k}$.

Show that, for every $x \in ]-\frac{1}{4}, \frac{1}{4}[$, $F(x) = 1 + x(F(x))^{2}$.

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