Questions tagged [bell-numbers]
For questions related to the Bell numbers, a sequence of natural numbers that occur in partitioning a finite set.
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Compute $S_n=\sum\limits_{a_1,a_2,\cdots,a_n=1}^\infty \frac{a_1a_2\cdots a_n}{(a_1+a_2+\cdots+a_n)!}$
It is tagged as an open problem in the book Fractional parts,series and integrals. If this proof is valid , I don't have any idea how to get it published so I posted it here .
$\displaystyle \sum_{...
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Extending Bell Numbers to Fractional Values
An identity of the Bell numbers is given by
$$B_n=\frac{1}{e}\sum_{x=1}^\infty \frac{x^n}{x!}$$
and I was wondering if it would be valid to define fractional Bell numbers in the same way, to preserve ...
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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 ...
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Sum of Bell Polynomials of the Second Kind
A problem of interest that has come up for me recently is solving the following.
$$\frac{d^{n}}{dt^{n}}e^{g(t)}$$
There is a formula for a general $n$-th order derivative of a composition as shown ...
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Partitions and Bell numbers
Let $F(n)$ be the number of all partitions of $[n]$ with no singleton blocks.
Find the recursive formula for the numbers $F(n)$ in terms of the numbers $F(i)$, with $i ≤ n − 1$
Find a formula for $F(...
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Taylor series of a power tower
I recently proved that the Taylor Series of $\exp(\exp(x))$ is given by
$$\exp(\exp(x))=\sum_{n=0}^\infty \frac{eB_n x^n}{n!}$$
where $B_n$ are the Bell Numbers.
However, I can't figure out a Taylor ...
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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.
$$a_n = \Big(\frac{1-q^{f \cdot n}}{1-q^n}\Big)\displaystyle\Big(1+\sum_{i=0}^{n-1}\binom{n}{i}p^{n-i}q^ia_i\Big)$$...
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Compact formula for the $n$th derivative of $f\left(\sqrt{x+1}\right)$?
I'm interested in a general formula for
$$\frac{d^n}{dx^n}\Big[f\left(\sqrt{x+1}\right)\Big].$$
In particular, Fàa di Bruno's formula gives
$$\frac{d^n}{dx^n}\Big[f\left(\sqrt{x+1}\right)\Big]=\sum_{k=...
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The number of partitions of $\{1, \ldots, n+1\}$ into subsets of nonconsecutive integers
For each integer $n$, let $a_n$ be the number of partitions of $\{1, \ldots, n+1\}$ into subsets of nonconsecutive integers.
I found (by listing) that $ a_1, a_2, a_3, a_4$ are $1, 2, 5, 15$ ...
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How are the Bell numbers related to this exponential series?
I recently started studying about the exponential series, and came across this infinite series $
{S}_{k}\mathrm{{=}}\mathop{\sum}\limits_{{n}\mathrm{{=}}{0}}\limits^{\mathrm{\infty}}{\frac{{n}^{k}}{n\...
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Is there a simple lower bound or approximation for the Bell numbers?
I'm not quite certain what descriptors to use to describe the solution I'm looking for, but is there an approximation or useful lower bound for the Bell numbers, for which the amount of terms used in ...
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Can you help with this proof that the $n$-th Bell number is bounded by $n!$ for all natural numbers $n$?
I am trying to prove that an upper bound for the nth Bell number is n factorial. I am trying to do this by induction. Firstly, the nth Bell number is given by:
$B_{n}=\sum\limits^{n-1}_{k=0} B_{k}{n-...
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$F(n)$ is number of ways to partition set of $n$ without singleton blocks. Prove that $B(n) = F(n) + F(n+1)$
In this case $B(n)$ is $n$-th Bell number.
To be honest, I would really love to know if there is a combinatorial proof for that. If there is not, other proofs are appreciated too.
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Bell Polynomials
The complete Bell polynomials $B_n(x_1, x_2, \ldots, x_n)$ are defined through the relation
$$\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}...
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On the ratio $\frac{F_n}{B_n}$
One of the interesting limits that I came up with is:
$$\lim_{n\to\infty} \frac{F_{n}}{B_{n}}\;\;\;\;\;\;\;\;\;\; \left( n \in \mathbb N^+\right)$$
Where $F_n$ is the nth Fibonacci number and $B_n$...
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