# Tagged Questions

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### Difficult generating function

Prove that the coefficient of $x^i$ for $i=1,2,3,...$ in the expansion of $\prod_{n=0}^{\infty} (1+x^{3^n})(1+x^{4^n})(1+x^{5^n})(1+x^{6^n})$ is greater than $0$.
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### Fermat Last theorem on Poly-Euler numbers

The poly-Euler numbers, denoted as $E_{n}^{(k)}$, are defined by the following generating functions :$${2\operatorname{Li}_k(1-e^{-x}) \over 1+e^{-x}}=\sum_{n=0}^\infty E_n^{(k)}{x^n\over n!}$$ The ...
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### Exponential generating function for number of 10 length sequences built from the alphabet, with some restrictions

I've got the following homework question. If anybody could possibly point me in the right direction, that would be great: Suppose X is a sequence with 10 terms built from 26 letters {a, b, c, ..., ...
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### Applications of generating functions to number theory

I am familiar (at least at a cursory level) with the extensive role generating functions play in the theory of partitions. What are some other prominent applications of generating functions to number ...
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### Proof that the series for the generating function of the partition function converges?

For $|q| < 1$, the generating function of the partition function $p(n)$ is given by $$\sum_{n=0}^\infty p(n) q^n = \prod_{k=1}^\infty {1 \over 1-q^k}. \tag{1}$$ I have an intuitive ...
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