# Tagged Questions

Generating functions are formed by making a series $\sum_{n\geq 0} a_n x^n$ out of a sequence $a_n$. They are used to count objects in enumerative combinatorics.

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### Is there a closed form expression for this sum involving Stirling number of second kind

The expression I am trying to simplify is the following: $$f(x)=\sum_{n\ge k}S(n,k)(L)_{n}x^n$$ where $S(k,n)$ is the Stirling number of second kind for $k,n\in \mathbb{Z}^+,\ L\ge n$ and $(L)_n$ is ...
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### Find the exponential generating function for the number of ways to distribute $r$ distinct objects into five different boxes

Find the exponential generating function for the number of ways to distribute $r$ distinct objects into five different boxes when $b_1<b_2\le 4$, where $b_1,b_2$ are the numbers of objects in boxes ...
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### Partial fraction of a generating function

I am solving a recurrence relation $a_0 = a_1 = a_2 = 1, a_{n+3} = a_{n+2} − 2a_{n+1} − 4a_n$ for $n \ge 0$ I got a generating function for this sequence $f(x) = \frac{2x^2+1}{4x^3+2x^2-x+1}$ ...
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I have three natural numbers $a, b, c$ such that $a + b + c = n$ and I'm looking for $\sum abc$. So far I've figured out that the generating function for $p(n,3)$ might be $\frac{x^3}{(1-x)(1-x^2)(1-... 1answer 18 views ### Trouble with integer partition proof I am reading Keller & Trotter: Applied Combinatorics, pg. 155, and I am having trouble with an intermediate step in a proof. The proof deals with integer partitions: And the part I can't ... 1answer 44 views ### A Finite Combinatorial Sum It can be proved by induction or telescoping sum that $$\sum_{i=0}^n {2i\choose i}\frac{1}{2^{2i}}=(2n+1){2n\choose n}\frac{1}{2^{2n}}.$$ However, without knowing the right hand side in advance, I ... 1answer 25 views ### Arrangements with no anomalous neighborhoods How many ways can$8$boys and$20$girls be ordered such that for each boy at position$i$, there is no neighborhood (of$2n+1$points with$n > 0$) consisting of positions$j \in [i-n,i+n]$that ... 1answer 58 views ### Find closed formula for$a_{n+1}=(n+1)a_{n}+n!a_{n+1}=(n+1)a_{n}+n!$where a0=0 and n>=0. To get the closed form, I'm trying to find an exponential generating function for the above recurrence, but it doesn't seem to be very nice. Am I going ... 1answer 58 views ### Find closed formula for the recurrence$a_{n}=na_{n-1}+n(n-1)a_{n-2}a_{n}=na_{n-1}+n(n-1)a_{n-2}$where a0 = 0, a1=1, and n >= 2. I found an exponential generating function for this recurrence, but cant seem to find the closed form because the generating function ... 1answer 28 views ### Proof gone wrong: Probability Generating functions. I'm trying to proof something but I'm getting a different answer than my textbook and I don't know where I've gone wrong. The question concerns a random discrete variable$Y$taking on values in$\...
Consider the inequality $x1+x2+x3+x4 ≤n$ where $x1, x2, x3, x4, n ≥ 0$ are all integers. Suppose also that $x2 ≥ 2$, $x3$ is a multiple of 4, and $1 ≤ x4 ≤ 3$. Let cn be the number of solutions of the ...