All Questions
Tagged with bell-numbers sequences-and-series
10
questions
2
votes
1
answer
68
views
Closed-form expression for the infinite sum in Dobiński's formula
In combinatorial mathematics, I learned about Dobiński's formula for the $n$-th Bell number $B_n$, which states that:
Dobiński's formula gives the $n$-th Bell number $B_n$ (i.e., the number of ...
- 2,292
0
votes
1
answer
108
views
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
1
vote
1
answer
85
views
Closed Form for Geometric-like Finite sum of Bell Polynomials
I'm trying to see if there's a nice closed form expression for the following sum:
$\sum_{k=0}^{M} \cos(\pi k t) B_k(x)$
where $M \in \mathbb{N}$, $t \in (0,1)$, and $x \in \mathbb{R}^+$.
Notation: ...
- 317
1
vote
1
answer
46
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How to show that $\sum_{l=0}^{\infty} \dfrac{(l+1)^{n-1}}{l!}=\sum_{l=0}^{\infty} \dfrac{l^{n}}{l!}$ (proof of Dobiński's formula)?
I am reading a proof of Dobiński's formula in Béla Bollabás book "The Art of Mathematics" (p. 144). There he uses
$$\frac{1}{e}\sum_{l=0}^{\infty} \dfrac{(l+1)^{n-1}}{l!}=\frac{1}{e}\sum_{l=...
- 889
4
votes
2
answers
1k
views
Can you help with this proof that the $n$-th Bell number is bounded by $n!$ for all natural numbers $n$?
I am trying to prove that an upper bound for the nth Bell number is n factorial. I am trying to do this by induction. Firstly, the nth Bell number is given by:
$B_{n}=\sum\limits^{n-1}_{k=0} B_{k}{n-...
- 131
0
votes
0
answers
143
views
On the period of Bell numbers modulo a prime
Like my recent question, this one is related to the conjecture stating that, for a prime $p$, the (exact) period of $n\mapsto B_n\bmod p$ is equal to $(p^p-1)/(p-1)$. That question covers a test ...
- 40.1k
3
votes
1
answer
256
views
What's the sum of the series $\sum\limits_{n\geq 0}\frac{n^x}{n!}$ with $x$ a positive real number?
By the ratio test the series
$$
\sum_{n\ge0}\frac{n^x}{n!}
$$
is convergent, but I know no method to evaluate it.
Since it's a convergent series then my question here is:
Is there a closed form ...
- 7,195
4
votes
1
answer
3k
views
Exponential Generating Function Stirling Numbers
In class we found the exponential generating function for the Bell numbers $B_n$ which are defined by the recurrence $B(0) = 1$, $B(1) = 1$ and $B(n+1) =\sum_{i=1}^n\dbinom{n}{i}B(n-i)$ for all$ n\geq ...
- 391
5
votes
2
answers
524
views
How are the Bell numbers related to this exponential series?
I recently started studying about the exponential series, and came across this infinite series $
{S}_{k}\mathrm{{=}}\mathop{\sum}\limits_{{n}\mathrm{{=}}{0}}\limits^{\mathrm{\infty}}{\frac{{n}^{k}}{n\...
- 195
8
votes
1
answer
763
views
Sum of Bell Polynomials of the Second Kind
A problem of interest that has come up for me recently is solving the following.
$$\frac{d^{n}}{dt^{n}}e^{g(t)}$$
There is a formula for a general $n$-th order derivative of a composition as shown ...
- 2,120