# Reference request: Gessel interview’s generating function identities

algebraic-combinatoricsco.combinatoricscombinatorial-identitiesgenerating-functionsgraph theory

In this interview, Ira Gessel mentions the following results:

Result 1: Let $$B_n$$ denote the $$n^{\text{th}}$$ Bernoulli number.
Define the series
$$B(x) = \sum_{n=2}^{\infty} \frac{B_nx^{n-1}}{n(n-1)}.$$
Let $$v_n$$ be the sequence of reals with
such that the series
$$F(x) = \sum_{n=1}^{\infty} \frac{v_nx^n}{n!}$$
satisfies
$$F\left(\sqrt{2x – 2(\log(1+x))}\right) = x.$$
Then we have
$$e^{B(x)} = \sum_{n=0}^{\infty} \frac{v_{2n+1}x^n}{2^n n!}.$$

Result 2: Let $$s_n$$ be the number of strongly connected, directed graphs with vertex set $$\{1, \dots, n\}$$. Let $$t_n$$ be the number of strongly connected tournaments with vertex set $$\{1, \dots, n\}$$. Let
$$T(x) = \sum_{n=1}^{\infty} \frac{2^{\binom{n}{2}}t_n x^n}{n!}.$$
Then
$$\log\left(\frac{1}{1-T(x)}\right) = \sum_{n=1}^{\infty} \frac{s_nx^n}{n!}.$$

Result 3: Let $$A_n(t)$$ denote the $$n^{\text{th}}$$ Eulerian polynomial. Then
$$\frac{A_n(t)}{(1-t)^{n+1}} = \sum_{k=0}^{\infty} k^nt^k.$$

My Question: What are the references for the proofs of results 1 through 3 above?