grandes-ecoles 2020 QIV.1

grandes-ecoles · France · x-ens-maths-d__mp Sequences and Series Functional Equations and Identities via Series

1. We denote $E_n = \operatorname{Card} \operatorname{MD}(n)$ and $\mathscr{I}_n$ the set of odd numbers in $\Delta_n$. a. Prove that for $n \geq 1$: $$E_{n+1} = \sum_{i \in \mathscr{I}_{n+1}} \binom{n}{i-1} E_{i-1} E_{n+1-i}$$ b. Deduce that for $n \geq 1$: $$2E_{n+1} = \sum_{i=0}^{n} \binom{n}{i} E_i E_{n-i}$$

1. We denote $E_n = \operatorname{Card} \operatorname{MD}(n)$ and $\mathscr{I}_n$ the set of odd numbers in $\Delta_n$.\\
a. Prove that for $n \geq 1$:
$$E_{n+1} = \sum_{i \in \mathscr{I}_{n+1}} \binom{n}{i-1} E_{i-1} E_{n+1-i}$$
b. Deduce that for $n \geq 1$:
$$2E_{n+1} = \sum_{i=0}^{n} \binom{n}{i} E_i E_{n-i}$$

Paper Questions

QI.1 QI.2 QI.3 QII.1 QII.2 QII.3 QII.4 QII.5 QII.6 QIII.1 QIII.2 QIII.3 QIII.4 QIII.5 QIV.1 QIV.2 QIV.3 QIV.4 QIV.5 QIV.6 QIV.7 QV.1