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?
Best Answer
Result 1 can be found in N. G. de Bruijn, Asymptotic Methods in Analysis, Dover, New York, 1981, p. 71.
Result 2 follows from a formula of E. M. Wright, though he didn't state it in this form. A discussion of this formula, with a generalization, can be found in Kassie Archer, Ira M. Gessel, Christina Graves, and Xuming Liang, Counting acyclic and strong digraphs by descents, Discrete Math. 343 (2020), 112041, 14 pp. https://doi.org/10.1016/j.disc.2020.112041. See Proposition 8. The arXiv version is https://doi.org/10.48550/arXiv.1909.01550.
As Sam and Richard noted, Result 3 is well known. It is equivalent to Euler's definition of the Eulerian polynomials. The combinatorial interpretation of the Eulerian polynomials is a special case of a much more general result of MacMahon, though I don't think that MacMahon recognized this special case as being noteworthy, nor did he connect it with Euler's work.