Questions tagged [closed-form]
A "closed form expression" is any representation of a mathematical expression in terms of "known" functions, "known" usually being replaced with "elementary".
3,990 questions
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Expected number of lattice points in a randomly inscribed triangle
Question. Let $C_R$ be the closed disk of radius $R$ centered at the origin. Let $T$ be a random triangle formed by three vertices $V_1, V_2, V_3$ chosen independently and uniformly from the ...
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Formula for number of unlabelled trees of n vertices [closed]
How do we prove that no closed-form expression exists for the number of non-isomorphic unlabeled trees of n vertices? How do we also prove that no closed-form expression can give the degree sequences(...
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Closed form for a symmetric sum of squared binomials
I am trying to find a closed form for the following sum: $$ \sum_{k=0}^{n-1} \left( \frac{1}{(k+1)(n-k)} \cdot \binom{n+1}{k+1}^2 \right) $$
What I have tried so far
I tried to simplify the expression ...
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Analytic sum of an alternating series$\sum\limits_{n=1}^{\infty}(-1)^{n} \frac{n}{\left(n+\sqrt{a+n^2}\right)^2}$
I recently came across the following series with a positive real number $a$:
\begin{align}
S(a) = \sum _{n=1}^{\infty}(-1)^{n} \frac{n}{\left(n+\sqrt{a+n^2}\right)^2}
\end{align}
Does anyone know if ...
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Counting unique-loop configurations on a $3 \times n$ grid
This is a smaller post that relates to these previous questions that I have asked $(1)$ $(2)$. Context for the (seemingly arbitrary) formulas may be found there.
Let us consider a $3 \times n$ grid ...
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Is this a correct closed form of $\sum_{n=0}^k(-1)^nn\binom kn$?
I have the sum
$$\sum_{n=0}^k(-1)^nn\binom kn$$
where I expect $k\ge 2.$ My analysis is that
$$(n+1)\binom k{n+1}-n\binom kn=k\binom{k-1}n$$
and therefore the original sum evaluates as
$$\sum_{n=0}^k(...
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Minimum number of forbidden cells for a unique loop on $m \times n$ grid
I recently asked about the minimum number of black (forbidden) squares needed to guarantee a unique "loop" cycle on an $n \times n$ grid. It seems natural, then, to consider $m \times n$ ...
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Closed form for Dirichlet series whose coefficients are the Möbius function times a geometric series
By definition,
$$
\sum_{n=1}^{+\infty}\frac{1}{n^s} = \zeta(s)
\tag{*}
$$
when the real part of $s$ is large enough ($>1$). I am also aware that
$$
\sum_{n=1}^{+\infty}\frac{\mu(n)}{n^s} = \frac{1}...
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Calculation of the derangements (subfactorial, number of fixed-point-free permutations) from the normal factorial
How can I derive $\boxed{
!n = \left\lfloor \dfrac{n!+1}{e} \right\rfloor,~~~ n \ge 1
}~?$
It is $\boxed{
!n
=n! \displaystyle\sum\limits_{k=0}^{n} \dfrac{(-1)^k}{k!}
}$ (wikipedia).
So I tried ...
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Closed form solution for $ \int\limits_{0}^{1} x^{x} (1-x)^{1-x} dx $
Does anyone have any hints or ways forward (or even better, a solution) for this integral?
$$\int_0^1 x^x (1-x)^{1-x} dx $$
I've tried multiple contour's but none seem to work. The only path forward I'...
user1707952
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Calculating $\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}\frac{(-1)^{n+m}}{\sqrt{n+1}\,\sqrt{n+2m+2}}$
Does the following series:
$$\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}\frac{(-1)^{n+m}}{\sqrt{n+1}\,\sqrt{n+2m+2}}$$
have a closed form? If so, how to obtain it?
I tried to use the Laplace representation:...
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Minimum # of black squares to guarantee uniqueness of loop visiting all white squares
I believe context will help before the statement of the problem. I was asked
In the picture below, can you find a closed, non-intersecting loop visiting every white square exactly once? (If you do, ...
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$\text{erfc}$ Integral
Is there any closed form for $$I(m,n) = \int_{0}^{\infty} \text{erfc}^n(x^m) \mathrm dx \tag1$$
I was able to obtain a closed form when $n = 1$.
\begin{align}
I(m, 1) &= \int_{0}^{\infty} \text{...
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Closed form of $P = \prod\limits_{k=1}^{N} k^{k^2}$ [duplicate]
What is the closed form (in terms of $N$) of $\color{blue}{\mathbf{P = \prod\limits_{k=1}^{N} k^{k^2}}}$?
When I say closed form, it should not contain another $N$-term or Infinite term product or sum ...
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Closed form expression of a sequence
Two players are playing a game. Initially, a positive integer $n$ is written. Players take turns, starting with player 1, erasing the written integer $k$ and either write $k-1$ or $\frac{k}{2}$, as ...
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