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Here is a proof that $a_1(n, p, q) = a_2(n,p,q)$. The generating function for the Stirling numbers of the second kind is $$\sum_{n=i}^\infty {n\brace i}\frac{x^n}{n!}= \frac{(e^x-1)^i}{i!}.$$ Also $$\ …

I am interested in polynomials $G_n(z)$ defined by the recurrence $$G_{n+1}(z) - 2G_n(z) + (1-nz)G_{n-1}(z)=0$$ for $n\ge1$ with the initial values $G_0(z) = 1$ and $G_1(z) = 1$. The next few values a …

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 discu …

Let \begin{equation*} A(x,q) = e^{qx}A(x)=e^{(q+1)x+(e^x-1)^2/2} \end{equation*} and define $a(n,q)$ by \begin{equation*} A(x,q) = \sum_{n=0}^\infty a(n,q) \frac{x^n}{n!}. \end{equation*} Then $a(n,0) …

Let $$ A(x,q)=\sum_{i=0}^{\infty}\frac{x^i}{\prod\limits_{j=0}^{i}(1-f(q+j)x)}, $$ so that $A(x) = A(x,0)$. Define $a(n,q)$ by $$A(x,q) = \sum_{n=0}^\infty a(n,q) x^n,$$ so that $a(n) = a(n,0)$. I wil …

Here's a sketch of a proof, using Lagrange inversion, of the equation $(27x^2/4-1)F^3+3F-2=0$, where $$F(x) = \sum_{n=0}^\infty \binom{3n/2}{n}x^n.$$ One form of Lagrange inversion says that if $h(x) …

See Gilbert Labelle, On asymmetric structures, Discrete Math. 99 (1992), 141–164.

There are many ways to prove the formula $$ \sum_{n=1}^{\infty} \frac{n^{n-1}}{n!}(xe^{-x})^n = x.\tag{1}$$ As Alexandre Eremenko noted, one approach is Lagrange inversion. Another can be found at htt …

In my lecture slides that Tewodros cites, I studied symmetric functions that I called $g_m$, where $m$ is a positive integer, that have constant term 1 and satisfy $$ -L_m[g_m] = p_1,$$ where $L_m$ is …

If you multiply both sides by $e^{x+y}$ there's a simple bijective proof: In a nutshell, every 0-1 matrix consists of a matrix with a 1 in every row and column together with some all-zero rows and col …

Using Pietro Majer's bisection formula we find by a straightforward computation (I did it with Maple, but I'm sure it could be done without too much difficulty by hand) that the OP's formula for $$\su …

For Bender and Goldman's theory of prefabs, see http://www.iumj.indiana.edu/docs/20060/20060.asp. Michael Henle's theory of dissects is introduced in Dissection of generating functions, Studies in App …

These numbers count balanced signed graphs (without loops). A signed graph is a graph in which every edge has a sign, either positive or negative. It is balanced if every cycle has an even number of n …

The exponential generating function for permutations of order dividing $k$ is $$\exp\biggl(\sum_{d\mid k} \frac{x^d}{d}\biggr).$$ See, e.g., L. Moser and M. Wyman, On solutions of $x^d = 1$ in symmetr …

I started writing this before Richard's answer appeared, with which it overlaps a lot, but I still have something to add. Let us look at a more general problem: Suppose that $G(x) = 1+g_1x+g_2x^2+\cd …