Questions tagged [eulerian-numbers]
For questions about the Eulerian numbers $A_{n,k}$, defined as the number of permutations in the symmetric group $S_n$ having $k$ descents. Not to be confused with Euler’s number $e$ or the Euler numbers $E_n$.
64
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Request bibliographic reference(s) for finite alternating sums with Eulerian numbers
I would like to know a bibliographic reference for a math formula. I found this formula on Wikipedia, but no reference is given.
It's the 2nd formula ($A(n,k)$ is an Eulerian number):
$$\sum_{k=0}^{n-...
- 1
3
votes
2
answers
168
views
Sum of the series $\sum_{k=0}^\infty k^ne^{-k}$ for a positive integer $n$
How can we calculate the sum of the series $\sum_{k=0}^\infty k^ne^{-k}$ for a positive integer $n$? I tried: $$\sum_{k=0}^\infty k^ne^{-k} =(-1)^n \sum_{k=0}^\infty \frac{d^n}{dy^n}\bigg|_{y=1}e^{-ky}...
- 1,802
1
vote
1
answer
107
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Identity connecting Stirling numbers of both kinds with second order Eulerian numbers
Setting. On page 270 of the great book Concrete Mathematics by Graham, Knuth and Patashnik, the second order Eulerian numbers $\langle\!\langle {n\atop k}\rangle\!\rangle$ with $n,k\in\mathbb{Z}$, $n \...
- 23.1k
1
vote
2
answers
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Does $\sum_{k=0}^{\infty} \sum_{n=1}^{\infty} x^nn^k =cx-x\ln(x)+x\ln(1-x) $ for $0 < x < 1$ for some real $c$? If so, what is $c$?
Let
$g_k(x)
=\sum_{n=1}^{\infty} x^nn^k
$
and
$G(x)
=\sum_{k=0}^{\infty} g_k(x)
=\sum_{k=0}^{\infty} \sum_{n=1}^{\infty} x^nn^k
$.
For $0 < x < 1$
is
$G(x)
=cx-x\ln(x)+x\ln(1-x)
$
for some real $...
- 108k
6
votes
1
answer
214
views
The relation of the Bernoulli numbers to the Catalan numbers
The Bernoulli numbers $B_n$ are the backbone of calculus, and according to B. Mazur, they "act as a unifying force, holding together seemingly disparate fields of mathematics."
The Catalan ...
- 905
2
votes
0
answers
99
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Understanding the recurrence relation $f(n,k) = kf(n-1, k) + (n-k+1)f(n-1,k-1)$ for the Euler numbers $f(n,k)$
The Euler numbers $f(n,k)$ are given in Generatingfunctionology as the number of permutations of $[n]$ with exactly $k$ increasing runs (exercise 1.18.c). Its recurrence relation is
$$f(n,k) = kf(n-1, ...
- 402
0
votes
2
answers
148
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Proof that $e^{i\theta}/e^ {i\phi} = e^{i(\theta - \phi)}$ [closed]
Could anyone please help me with proving that
\begin{equation}
\frac{e^{i\theta}}{e^{i\phi}}= e^{i(\theta-\phi)}
\end{equation}
Is $e^{i\theta} = \cos(\theta)+i\sin(\theta)$ useful in this proof?
...
- 3
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vote
3
answers
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Power series of $x/(1-ae^{-x})$.
I am looking for a power series expansion for the function $x(1-ae^{-x})^{-1}$ (perhaps for $0<a<1$).
Using the Bernoulli numbers, I can write
\begin{align*}
\frac{x}{1-ae^{-x}} &= \frac{x}{...
- 389
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votes
0
answers
78
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Closed form of Eulerian numbers Proof
I was reading the proof of the closed-form formula of Eulerian numbers ($A(n,k)$) from Bona's book. I had a doubt in his classification of "removable fences" and how they eliminate the ...
- 13
1
vote
1
answer
101
views
Connecting Euler Numbers of the Second Kind and Unsigned Stirling Numbers of the First Kind. [closed]
The notation $ \left\langle\left\langle n\atop m\right\rangle\right\rangle$ denotes the Eulerian numbers of the second kind or order. Similarly, ${n \brack m}$ denotes the unsigned Stirling numbers of ...
- 372
-1
votes
1
answer
304
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Worpitzky's identity eulerian number - formula demonstration [closed]
Can someone tell me please how am I supposed to demonstrate the Worpitzky's identity :
$$
x^{n}=\sum_{k}\left\langle\begin{array}{l}
n \\
k
\end{array}\right\rangle\left(\begin{array}{c}
x+k \\
n
\end{...
8
votes
1
answer
299
views
Bounds on the difference between the polylogarithm with negative base and the gamma function
Trying to understand intuitively the Gamma function I started to think of it as a way to measure how much each factorial power "helps" $x^n$ in the infinite sum of $e^x$, thus trying to ...
5
votes
0
answers
71
views
Connection between Eulerian numbers and number of elements in set of uniform variables greater than the mean?
I was recently investigating the following question: Given $n$ independent $\text{Unif(0, 1)}$ variables $U_1,\ldots,U_n$, let $m$ be the number of elements in $[U_1,\ldots,U_n]$ that are greater than ...
3
votes
0
answers
62
views
The second-order Eulerian numbers meet the Clausen numbers
The (generalized) Clausen numbers A160014 are defined as
$$\operatorname{C}_{n, k} = \prod_{ p\, -\, k\, |\, n} p \quad (p \in \mathbb{P})$$ where $\mathbb{P}$ denotes the primes.
The classical ...
- 905
6
votes
2
answers
143
views
An identity related to the second-order Eulerian numbers.
Recently, some of the remarkable properties of second-order
Eulerian numbers $ \left\langle\!\!\left\langle n\atop k\right\rangle\!\!\right\rangle$ A340556 have been proved on MSE [ a ,
b , c ].
...
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