All Questions
Tagged with bell-numbers generating-functions
8
questions
0
votes
1
answer
108
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Stirling Numbers Exponential Generating Function Induction
I was reading the solution to a question written here, and it uses a fact which can be proved by induction.
The question is to show that an EGF for Stirling Numbers of the second kind with fixed $k$, ...
- 5
8
votes
1
answer
311
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Strange polynomial analog of the Bell numbers
Let $\vec{x} = (x_0, x_1, x_2, \dots)$ and $\vec{y}=(y_1,y_2,y_3, \dots)$ be two systems of parameters/variables. The Motzkin
polynomials $P_n(\vec{x},\vec{y})$ for $n \geq 0$ are defined by the ...
- 567
4
votes
1
answer
308
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Bell Polynomials
The complete Bell polynomials $B_n(x_1, x_2, \ldots, x_n)$ are defined through the relation
$$\sum_{n=0}^{\infty} B_n(x_1, x_2, \ldots, x_n) \frac{t^n}{n!} =\exp\Big( \sum_{n=1}^{\infty} x_n \frac{t^n}...
- 551
2
votes
1
answer
232
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Exponential generating function with Stirling numbers
I want to prove in particular this result-
$$
\newcommand{\gkpSII}[2]{{\genfrac{\lbrace}{\rbrace}{0pt}{}{#1}{#2}}}
\sum_{k \geq 0} \gkpSII{2k}{j} \frac{\log(q)^k}{k!} =
\frac{1}{\sqrt{2\pi}} \...
- 63
1
vote
2
answers
185
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An approximation of the ordered Bell numbers
So my problem is the following: I have $n$ ice-cream flavors and I must rank them, allowing that I can place more than one flavor in some ranks. So for example if I have 4 flavors, I can put in the ...
2
votes
1
answer
570
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Generating function for the number of partitions of [n] without singletons.
I know the generating function for the total number of partitions of [n] is
given by $$ B(x)=e^{e^x-1}$$ I am struggling to find $V(X)$, the exponential generating function for the number of ...
- 43
7
votes
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Complicated recursion formula, seems similar to Bell numbers?
I came up with a recursive formula for a problem I was working on. It is as follows.
$$a_n = \Big(\frac{1-q^{f \cdot n}}{1-q^n}\Big)\displaystyle\Big(1+\sum_{i=0}^{n-1}\binom{n}{i}p^{n-i}q^ia_i\Big)$$...
0
votes
2
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210
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Remebering bell numbers
I read about Bell numbers and I'm looking for a way to generate these numbers quickly for tests and exams. I know there is a recursively relation but it is not useful for big numbers .
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