# Tagged Questions

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In the course of working out the Maclaurin expansions of $e^{-x^2}$ and $cos(x^2)$, I ran into the following nested sum: $$\underbrace{ \sum_{a=0}^1 \left( a \sum_{b=0}^{a+1} b \left( ... 1answer 51 views ### Number of product pairs equal to or less than a number I would like to figure out how many ways there are to create product pairs equal to or less than a certain number. In other words, find a pair of integers (n,m) such that nm \le N for a given ... 3answers 84 views ### Product of “reversed” numbers Consider any 2 binary numbers, e.g.: 10101011 ; 11111101 and their product, say P. "Reverse" (mirror image) all the digits of the 2 numbers, e.g.: ... 1answer 165 views ### Partition Proof Let \lambda be a partition of N of rank r. How can I show that:$$\sum_wx^\lambda(w)=f^\lambda(-1)^{t(\lambda)}\prod^r_{i=1}(\lambda_i-1)!(\lambda'_i-1)!$$where w ranges over all ... 1answer 100 views ### Expression for sum of k-products of n variables Given n variables there are n \choose k different terms that are the product of k different variables. For example, in the case that n = 3, the k-products of the variables x_1, x_2, x_3, ... 1answer 98 views ### Identity involving a recursive product Here is yet another problem related to plane partitions. Given the recursive formula$$ \begin{align*} F(0)&=1,\\ F(r)&=\prod_{i=1}^r\frac{i+2r-1}{2i+r-2}F(r-1). \end{align*} $$How can we ... 1answer 59 views ### Proving a poset is atomic A poset (X,\le)  is atomic if it has both a smallest and largest element, it is graded ,and every element x of X is the join x_1\vee \dots\vee x_n of some elements of X (also written as ... 2answers 201 views ### Showing an indentity with a cyclic sum Let n\geqslant2, and k\in \mathbb{N} Let z_1,z_2,..,z_n be distinct complex numbers Prove that$$ \sum_{i=1}^{n}\frac {{z}_{i}^{n-1+k}} { \prod \limits_{\substack{j = 1\\j ...
The following is taken from page 8 of Alon and Spencer's The Probabilistic Method. $$\prod_{i = 0}^{n-1} \frac{v - 2i}{v-i} \sim e^{-n^2/2v}$$ as long as $v \gg n^{3/2}$, estimating ...
### the limit of infinite product $(1+y)(1+y^2)(1+y^3)(1+y^4)\cdots$
I wonder if the function $(1+y)(1+y^2)(1+y^3)(1+y^4)\cdots, 0< y<1$, converges to some well-known function. If we let \$ (1+y)(1+y^2)(1+y^3)(1+y^4)\cdots = \prod_{i=1}^\infty (1+y^i) = ...