Bourbaki

Exercise 5, § 7

Algebra, I, p. 143

Let XX be a set with one element. If nn is an integer ≥1\geq 1, denote by unu_n the number of elements of M(X)M(X) of length nn. a) For every set YY, now prove that one has

Card⁡(Mn(Y))=un⋅Card⁡(Y)n. \operatorname{Card}(M_n(Y)) = u_n \cdot \operatorname{Card}(Y)^n.
(Use exerc. 3.) b) Establish the relation
un=∑p=1n−1upun−pfor n≥2. u_n = \sum_{p=1}^{n-1} u_p u_{n-p} \quad \text{for } \quad n \geq 2.

c) \* Let f(T)f(T) be the formal series ∑n=1∞unTn\sum_{n=1}^{\infty} u_n T^n. Now prove that one has f(T)=T+f(T)2f(T) = T + f(T)^2. Deduce the formulas

f(T)=12(1−(1−4T)1/2),un=2n−1n!1.3.5...(2n−3)for n≥2.∗ f(T) = \frac{1}{2}(1 - (1 - 4T)^{1/2}), \qquad u_n = \frac{2^{n-1}}{n!} 1.3.5...(2n-3) \quad \text{for } n \geq 2.*
Find this last result again using exerc. 11 of E, III, p. 85.

Solution

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Machine translation of the English, by gpt-5-mini. Not checked by a person. That model is a cut down version of the one the rest of this language was written by.