Questions tagged [closed-form-expressions]
For questions that specifically ask for determining a closed form of equations, integrals etc.
226
questions
1
vote
1
answer
46
views
Simplest way to generate integer coefficients with row sums equal to the terms of an arbitrary given sequence
Let $f(n)$ be an arbitrary function.
Let $\operatorname{wt}(n)$ be A000120 (i.e., number of ones in the binary expansion of $n$). Here
$$
\operatorname{wt}(2n+1) = \operatorname{wt}(n) + 1, \\
\...
- 4,044
15
votes
1
answer
689
views
New series for $\pi$ from string theory
This is a direct followup to the post Possible new series for $\pi$ by Timothy Chow
and its numerous answers and comments.
Using another formula in the same string theory paper by Saha and Sinha one ...
- 12.8k
0
votes
0
answers
59
views
Closed form for the A357990 using A329369 and generalised A373183
Let
$$
\ell(n) = \left\lfloor\log_2 n\right\rfloor, \\
\ell(0) = -1
$$
Let
$$
f(n) = \ell(n) - \ell(n-2^{\ell(n)}) - 1
$$
Here $f(n)$ is A290255.
Let $A(n,k)$ be a square array such that
$$
A(n,k)...
- 4,044
7
votes
1
answer
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views
Closed form for $\sum_{n=0}^\infty \frac1{2^{2^n}}$?
Is the sum of series $\displaystyle \sum_{n=0}^\infty \frac1{2^{2^n}} = \frac12 + \frac14 + \frac1{16} + \frac1{256} + \frac1{65536} + \dotsb \approx 0.8164215090218931$ representable in a closed form?...
- 79
6
votes
0
answers
737
views
For all $n\in \mathbb{N}$, How to find $\min\{ m+k\}$ such that $ \binom{m}{k}=n$?
I asked this question on MSE here.
Most numbers in pascal triangle appear only once (excluding the duplicates in the same row of the Pascal's triangle) but certain numbers appear multiple times. ...
- 431
3
votes
0
answers
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views
Sequence that sums up to A014307
Let $s(n,k)$ be a (signed) Stirling number of the first kind.
Let $n \brace k$ be a Stirling number of the second kind.
Let $a(n)$ be A014307. Here
$$
A(x) = \sum\limits_{k=0}^{\infty} \frac{a(k)}{...
- 4,044
2
votes
1
answer
161
views
Simplification of the closed form for the A329369
Let $s(n,k)$ be a (signed) Stirling number of the first kind.
Let ${n \brace k}$ be a Stirling number of the second kind.
Let
$$
f(n,m,i) = (-1)^{m-i+1}\sum\limits_{j=i}^{m+1}j^n s(j,i) {m+1 \brace ...
- 4,044
1
vote
0
answers
80
views
Closed form for the family of polynomials
Let $s(n,k)$ be a (signed) Stirling number of the first kind.
Let $R(n,x)$ be the family of polynomials such that
$$
R(2n+1,x) = xR(n,x), \\
R(2n,x) = x(R(n,x+1) - R(n, x)), \\
R(0, x) = x
$$
Let $\...
- 4,044
1
vote
1
answer
102
views
Closed formula / asymptotics for a generating function involving Gegenbauer / ultraspherical polynomials
Are there asymptotics, or even a closed form, for the following series
$$ \sum_{k = 0}^\infty e^{2 \pi i \sqrt{k^2 + (d-1) k} } \left( \binom{d+k}{k} - \binom{d+k-2}{k-2} \right) G_{\frac{d-1}{2},k}(t)...
- 353
1
vote
1
answer
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views
An "almost" geodesic dome
A regular $ n$-gon is inscribed in the unit circle centered in $0$.
We want to build an "almost" geodesic dome upon it this way: on each side of the $n$-gon we build an equilateral triangle ...
- 367
0
votes
0
answers
95
views
integral of exponential of Fourier series
I have encountered the following integral:
\begin{equation}
\int_0^{1} e^{-i F(x)} dx, \quad F(x) = \sum_{k=1}^L a_k \sin(2\pi k x) + b_k \cos(2\pi k x).
\end{equation} I have found several great ...
- 9
1
vote
0
answers
110
views
Representing A329369 using A358612
Let $a(n)$ be A329369 (i.e., number of permutations of $\{1,2,\dotsc,m\}$ with excedance set constructed by taking $m-i$ ($0 < i < m$) if $b(i-1) = 1$ where $b(k)b(k-1)\cdots b(1)b(0)$ ($0 \...
- 4,044
0
votes
1
answer
121
views
Is there a closed form for the integral $\int_{0}^{\infty}\ln(z)z^{\lambda - 1}\exp\left(-\frac{w}{2}\left(z + \frac{1}{z}\right)\right)\mathrm{d}z$?
As the title says, I would like to know if there is a closed form for the integral:
\begin{align*}
\int_{0}^{\infty}\ln(z)z^{\lambda - 1}\exp\left(-\frac{w}{2}\left(z + \frac{1}{z}\right)\right)\...
- 49
4
votes
1
answer
135
views
Closed form for the A110501 (unsigned Genocchi numbers (of first kind) of even index)
Let $a(n)$ be A110501 (i.e., unsigned Genocchi numbers (of first kind) of even index). Here
$$
a(n) = \sum\limits_{i=1}^{\left\lfloor\frac{n}{2}\right\rfloor}\binom{n}{2i}a(n-i)(-1)^{i-1}, \\
a(1) = 1
...
- 4,044
1
vote
0
answers
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Closed form for $ \sum_{a_1=0}^\infty~\sum_{a_2=0}^\infty~\cdots~\sum_{a_n=0}^\infty \dfrac1{(a_1!+a_2!+\ldots+a_n!)} $
I already posted this question on here.
After reading this post and the general solution for that case, I wondered if there is a closed form for the general solution for this sum:
$ \sum_{a_1=0}^\...
- 367