grandes-ecoles 2017 QIIB

grandes-ecoles · France · centrale-maths1__pc Sequences and Series Functional Equations and Identities via Series

We set for every integer $n \geqslant 0$, $$B _ { n } = \sum _ { k = 0 } ^ { n } S ( n , k )$$ where $S(n,k)$ denotes the number of partitions of $\llbracket 1, n \rrbracket$ into $k$ parts.
Prove the formula $$\forall n \in \mathbb { N } , \quad B _ { n + 1 } = \sum _ { k = 0 } ^ { n } \binom { n } { k } B _ { k }$$

We set for every integer $n \geqslant 0$,
$$B _ { n } = \sum _ { k = 0 } ^ { n } S ( n , k )$$
where $S(n,k)$ denotes the number of partitions of $\llbracket 1, n \rrbracket$ into $k$ parts.

Prove the formula
$$\forall n \in \mathbb { N } , \quad B _ { n + 1 } = \sum _ { k = 0 } ^ { n } \binom { n } { k } B _ { k }$$

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