# Tagged Questions

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### Ordinary Generating Function for Bell Numbers

In the OEIS entry for Bell numbers, there appears a generating function $$\sum_{k=0}^\infty B_k t^k = \sum_{r=0}^\infty \prod_{i=1}^r \frac{t}{1-it}$$ However, I could not locate any proof of ...
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### Approaches to implicitly defining generating function

First,every language in Chomsky hierarchy(or c.e.language) corresponds to a generating function,the set of the functions is GF,now,a question : is every generating function with integral coefficient ...
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### Are there generating functions of rational or integral solutions of Diophantine equation that

As we know,there are generating functions for c.e.languages which are some retricted rational or algebraic or transcendental functions dependent on the class of languages like regular ...
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### Is there inverse FFT algorithm for Fourier transform of a integer-valued random variable?

In many applications, it is possible to derive an explicit expression for the Fourier transform of a random variable $X$ $$\varphi (\theta ) = \sum\limits_{n = 0}^\infty {{p_n}} {e^{in\theta }}$$ ...
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### Generating function of factorable binary words

A word $w$ on the alphabet $A := \{0, 1\}$ is factorable if $$w = u^k \mbox{ where } u \in A^* \mbox{ and } k \geq 2.$$ Let $L$ be the language of the set of ...
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### Generating series of free PROs

Let $$G := \biguplus_{p \geq 0} \: \biguplus_{q \geq 0} G(p, q)$$ be a bigraded set of generators and $\mathcal{F}(G)$ be the free PRO generated by $G$ (see [1] for a net ...
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### Partitions-Sum of divisors identity

A few years ago I first read about the marvelous Euler identity: $\sum_{n\in\mathbb{N}}p(n)z^n=\prod_{k\geq1}\frac{1}{1-z^k}$, where $p(n)$ is the number of partitions of $n$ ($p(0)=1$ by ...
It is shown here on Mathworld's page on Stirling number of the second kind that $$\sum_{k=1}^n S(n,k) (k-1)! z^k = (-1)^n \text{Li}_{1-n}(1+\frac{1}{z})$$ where $S(n,k)$ is Stirling number of the ...
### The functional equation $f(f(x))=x+f(x)^2$
I'd like to gather information and references on the following functional equation for power series $$f(f(x))=x+f(x)^2,$$$$f(x)=\sum_{k=1}^\infty c_k x^k$$ (so $c_0=0$ is imposed). First things that ...