# Questions tagged [recurrences]

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### Bounding the $n$-th term of a sequence, given a non-linear recursive bound

I asked the following question in MSE: Let $a,b\in\mathbb{R}^+$. Suppose that $\{x_n\}_{n=0}^\infty$ is a sequence satisfying $$|x_n|\leq a|x_{n-1}|+b|x_{n-1}|^2,$$ for all $n\in\mathbb{N}$. How can ...
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### Reference request: recurrence relation for Catalan numbers

I would like to know if the following recurrence relation for Catalan numbers (see mathoverflow.net/questions/191524 and also math.stackexchange.com/questions/2113830) has appeared in a paper or a ...
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### Asymptotic analysis of $x_{n+1} = \frac{x_n}{n^2} + \frac{n^2}{x_n} + 2$

Problem: Let $x_1 = 1$ and $x_{n+1} = \frac{x_n}{n^2} + \frac{n^2}{x_n} + 2, \ n\ge 1$. Find the third term in the asymptotic expansion of $x_n$. I have posted it in MSE six months ago without ...
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### Two-term recurrence relation

We consider the following system of recurrence relations for $n \in \mathbb Z$ and $\vert \lambda \vert=1$ with $\lambda \in \mathbb{C}$ $$a_{n+1} = \lambda a_{n-1}+ \lambda^* a_n + \lambda^* n b_n$$ ...
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### What is the direct proof of the recurrence relation about lattice path enumeration given by Bizley?

Let $k$ be a nonnegative integer and let $m,n$ be coprime positive integers. Let $\phi_k$ be the number of lattice paths from $(0,0)$ to $(km,kn)$ with steps $(0,1)$ and $(1,0)$ that are never rising ...
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### Recurrence relation in matrices

I have the following recursive sequence: $Z_k = Z_{k-1} - AA^TZ_{k-1}xx^T$ where $Z_k \in \mathbb R^{n \times d}, A \in \mathbb R^{n \times d}, d > n, rank(A) = n, x \in \mathbb R^{d \times 1}$ $A$ ...
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### The number of permutations of given order

I want to count the number of permutations of the given order $k$ in $S_n\;(\sigma^k=id,\sigma^l\neq id\;for\;l<k)$. I found some works about that problem, but they are more general than necessary. ...
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