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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 …

As David noted, since the summands aren't in general integers, it's difficult to give a combinatorial interpretation to the formula. However, if we multiply by $a$ or $b$ we get integers and we can gi …

Formal power series with radius of convergence 0 often arise in counting labeled graphs. For example, the exponential generating function for labeled connected graphs is $\log G(x)$, where $$G(x) = \s …

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 …

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 …

The coefficient of $z^n$ is $$\sum_{0\le k\le n/2} (-1)^k \frac{k+1}{2n-3k+1}\binom{2n-3k+1}{n-2k}.$$ To see this, let $C(z)$ be the Catalan number generating function, $$C(z) = \frac{1-\sqrt{1-4z}} …

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 …

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 …

Here's a sketch of a derivation of (3) from (1). It's fairly straightforward to compute $$\sum_{n=0}^\infty F_n(x) z^n = \frac{1-2z-xz}{(1-xz)^2 -4z}.$$ If you expand this in powers of $z$ you get (3) …

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 …

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

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 …

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'm not completely sure what the problem is, but $$(1-x-y) c(x,y) = 1 - y C(xy) = 1 - \frac{1-\sqrt{1-4xy}}{2x},$$ where $C(z)$ is the Catalan number generating function, $$C(z) =\sum_{n=0}^\infty C_ …

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 …