Questions tagged [factorial]
Questions on the factorial function, $n!=n\cdot(n-1)\cdot...\cdot1$. Consider using the tag (gamma-function) if dealing with noninteger arguments.
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Factorials dividing sums of factorials
For the case of a factorial dividing another factorial, one has that $m! \mid n!$ if and only if $m \le n$, for any two positive integers $m$ and $n$ (yes, they must be strictly positive, because ...
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When does a line equal the Gamma function?
How would you solve for $x$ in the following equation:
$$
x = \left( x - 1 \right) ! = \int_0^\infty t^{x-1} e^{-t} dt
$$
If we are only concerned about integers, then clearly, the only solution is $1$...
-2
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0
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Proving $\frac{(n^2)!}{(n!)^{n+1}}$ is an integer using number theory [duplicate]
This is an intermediate result for an AMC problem: 2019 AMC10A Problem 25. The solution presents two ways to determine that $\frac{(n^2)!}{(n!)^{n+1}}$ is an integer. I understand the combinatorial ...
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2
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How would you go about finding the exponent for any given x that most closely matches or surpasses factorial growth?
While analyzing the factorial function and comparing it to basic exponentiation, I couldn't help but notice the obvious fact that exponentiation can eventually overtake factorialization if the ...
2
votes
1
answer
73
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Sum of Hermite polynomials
I am trying to find a closed form (or the tightest upper bound possible) for the following sum:
$$
\sum_{n=0}^{\infty} \frac{H_{2n}(x)w^n}{(2n+1)!}
$$
From the equalities $
\sum_{n=0}^{\infty} \frac{...
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Counting Paths in the XY Plane (Discrete math) [duplicate]
I need help with the following mathematical task:
A particle moves in the xy-plane according to the following rules:
U: (m, n) → (m+1, n+1)
L: (m, n) → (m+1, n-1)
where m and n are integers. I need ...
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1
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Is the factorial of any number equal to zero?
Was playing around on desmos and discovered that the graph of x! = y does not interact with zero at any point. Was wondering if any maybe complex number or number that desmos didn't properly compute ...
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Two different expressions from solving $s(n, m) = s(n-1, m) + s(n-1, m-1)$, alternate proof they are equal.
Before reading this please quickly look at this question I asked before, and the accepted answer. Upon further inspection, I found another solution to the recurrence relation, namely (the reciprocal ...
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Ratio of a factorial is Integer [duplicate]
I had been given a question
$ (15(n!)^2 + 1)/(2n-3) = l $
Find sum of values of n for which l is integer
Answer is given to be 90 but by trial I find n=1 and 2 to be only solution.
I think that if n ...
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2
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Prove a certain equation [duplicate]
Can anyone prove this equation? $$\sum_{n=0}^{\infty} (-1)^n \left( \frac{(2n-1)!!}{(2n)!!} \right)^3 = \left( \frac{\Gamma\left(\frac{9}{8}\right)}{\Gamma\left(\frac{5}{4}\right) \Gamma\left(\frac{7}{...
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answer
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Show that there are infinitely many $a\in\mathbb{N}$ such that $a!+(a+2)!$ divides $(a+2\lfloor{\sqrt{a}}\rfloor)!$.
The question is:
Show that there are infinitely many natural number $a$ such that $a!+(a+2)!$ divides $(a+2\lfloor{\sqrt{a}\rfloor})!$.
My attempt:
I let $a=(b-1)^2$ for a natural number $b>1$. ...
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Seeking Clarification on Proof for P(n, n) = P(n, n-1) Equality
I've encountered a mathematical challenge that has me scratching my head, and I'm hoping to get some assistance in validating my solution. The problem revolves around proving the equality of P(n, n) ...
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A nested double sum(to do with e?)
I’ve come across this nested double sum while doing an investigation but cannot seem to find a closed form for it.
$$1+\sum_{i=1}^\infty{\frac{1}{i!} \sum_{j=0}^i}{\frac{1}{j!}}$$
This is about the ...
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Approximation of gamma function via Riemann sums at integer points
I found something curious. We know that the gamma function is defined as
$$ \Gamma(n+1) := \int_{t=0}^\infty t^n \exp(-t) dt,$$
and it has the property that $\Gamma(n+1) = n!$ for non-negative integer ...
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Why is the following limit not 1?
Consider $$\lim_{n\to\infty} f(n)=\lim_{n\to\infty}\frac{(n!)!}{(n!-n)!}\tag{1}$$
For large $n$, one can ignore $n$ wrt. $n!$ in the denominator. The limiting value should therefore approach $1$. ...
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Completing the summation equality with the example $\sum_{n=2}^{\infty} \sum_{m=1}^n \frac{\prod_{i=0}^n(i+2)}{n+m}$
I have the following summation that when expanded represents the factorial, for example:
$$\sum_{n=2}^{\infty} \sum_{m=1}^n \frac{\prod_{i=0}^n(i+2)}{n+m}=(2\cdot4)+(2\cdot3\cdot5)+(2\cdot3\cdot4\...
- 243
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Find the value or prove that it's impossible [duplicate]
Let there be 4 integers $w,x,y,n$ belonging to $ℤ+$ such that $w,x,y \geq n$ and $n \gt 2$. Find the value of $w$ in terms of $x, y, n$ or prove that it's impossible if $ x \neq y $ and
$\frac{w!}{(w-...
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Correct approach on log(n+m+z)!
So I have for simpler notation, for a fixed infinitely large natural number $n$ and all finite natural numbers $m,z$, then $L(n+m)=\log(n+m)!$, and this is equal to $$L(n+m)=\log n!+\sum_{k=1}^m\log(n+...
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Removing first factor of q-pochammer
What does it mean to say 'removing the first factor ' from the following function :
$$f(a)=\int_0^{\infty}t^{x-1}\frac{(-at;q)_{\infty}dt}{(-t;q)_{\infty}}$$
This interval converges when $x>0$ and $...
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2
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Solve factorial equal to product of other factorials, $a! = b! \times 5! \times 3!$
General equations of factorials equal to the product of other factorials, e.g. $a! = b! \times c! \times d!$, have been asked before on this site and turn out to be an open problem, though only four ...
- 1,548
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Evaluate $\lim_{n\to\infty}\frac{12^n\cdot(n!)^5}{(n^2+n)^{2n}\cdot(n+1)^n}$
Evaluate the following limit: $$\lim_{n\to\infty}\frac{12^n\cdot(n!)^5}{(n^2+n)^{2n}\cdot(n+1)^n}$$
The factorial function is creating a problem for me here. I can manage all other terms (by clubbing ...
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1
answer
30
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Calculate the combinations of 4 items in different number of sets [duplicate]
Let's say I have four colors. Red, Green, Blue, Black
I want to find all different combinations that those can be put to, being able to use sets of 4 colors, down to 1 color. Order will always be the ...
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How can i prove this sum of factorials?
i was messing around whith some trig identities and i came across this equation:
$$\sum_{k=0}^n\frac{1}{(2k)!(2(n-k)+1)!}=\frac{1}{2}\frac{2^{2n+1}}{(2n+1)!}\quad\quad(1)$$
This formula becomes pretty ...
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Can $1!+2!+3!+...n!$(written in base 18) be a perfect square in decimal? [closed]
Can $1!+2!+3!+...n!$(written in base 18) be a perfect square in decimal?
I know that $1!+2!+3!+...n!$ is never a perfect square if $n\geq5$, since the last digit of the sum is $3$, but I don't know if ...
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Proof that if $p^r$ divides $\binom{2n}{n}$ then $p^r\le2n$. [duplicate]
Let $p$ be a prime and let $n$ be a positive integer. Then $p$ divides $\binom{2n}{n}$ exactly $\sum_{i=1}^{\lfloor \log_p 2n\rfloor} \left\lfloor\frac{2n}{p^i}\right\rfloor-2\left\lfloor\frac{n}{p^i}\...
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Integrating a multidimensional minimum function?
Back some time ago, a friend challenged me to find this integral
$$I = \int_0^1\int_0^1\cdots\int_0^1\min(x_1+x_2+\cdots+x_n, 1)\text{ d}x_1\text{ d}x_2\cdots\text{ d}x_n$$
I started off by trying to ...
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Proof by induction of infinite product
I just started learning inductive proofs but I am stuck trying to proof that the following equation applies for all $n\in\mathbb{N}_0$.
\begin{align}
n!&=\prod_{i=1}^{\infty}{\left[\left(\frac{i+1}...
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Why are these two sums with factorials equal?
An exercise from Grimmett & Stirzaker 3.1 question 2: For a random variable $X$ with a mass function on positive integers $$f(x)=\frac{1}{e^2-1}\frac{2^x}{x!},$$ what is the probability that $X$ ...
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Confusion in factorials/permutations.
Assume we have 8 elements arranged in two rows (A, B, C, D, a, b, c, d) it is known that the number of arrangements is 8! = 40320
What I thought about is:
Step 1: We assume the two rows initially have ...
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1
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Help me prove a fun identity of binomial coefficients [duplicate]
Let $M$ be a positive integer. Then I believe that the following fun identity is true:
$$\sum_{k=0}^{\text{Floor}(M/2)}2^{M-2k}{M\choose 2k}{2k\choose k} = {2M\choose M}$$
Numerically it checks out ...
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Subsets of $\mathbb{Z}$ satisfying the factorial property
Consider the subset $S \subseteq \mathbb{Z}$ given by:
$$S = \{ 2^i 3^j : i,j \ge 0 \}$$
Define the sequence $(a_k)$ to be the elements of $S$ in increasing order (with the standard order on $\mathbb{...
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Proving equality of factorial expressions
I was solving some combination problems and noticed that some of them can have two or more different ways to solve.
Look at these examples :
In how many ways can we wear a shoes and b shirts together:
...
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Alternating sum of products of half-shifted binomial coefficients
I have observed that the following identity seens to hold:
Let $k \geq 2$ and $0 \leq p, q \leq k+1$. Then
$$\sum_{m = 0}^{k+1-q} \begin{pmatrix}
-\frac12 + \lceil \frac{m+q}{2} \rceil ...
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What is the asymptotic version of the solid angle formula in $d$ dimensions?
It is well known that the solid angle in an euclidian space of $d$ dimensions ($d = 2 n$ or $d = 2 n + 1$, where $n = 1, 2, 3, \dots, \infty$) is given by these formulae:
\begin{align}\tag{1}
\Omega_{...
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Any good analytical approximation for the inverse factorial moment of Poisson random variable?
I have the following question:
Consider a random variable $M \sim \mathrm{Poisson} (\lambda)$. I would like to evaluate the following expectation:
\begin{align*}
\mathbb{E} \left[ 1 / \Gamma (M + a) \...
3
votes
1
answer
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Closed form solution for indexing combinations from n choose r
I have a list of combinations resulting from n choose r where order doesn't matter and without repeats, and the list is ordered based on the first choice. To make this concrete, in my specific case, ...
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Are there infinite triplets that satisfy a! = b!⋅c! where a ≠ b+1? [duplicate]
I'm not a mathematician. I was recently watching a YT math question on what is 10! ÷ 6! = x!. It got me thinking, are there infinite triplets (a, b, c) that satisfy a!= b!⋅c!?
I wrote a short python ...
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A number-theoretic proof that $(n^2)!/(n!)^{n+1}$ is an integer [closed]
I have seen a number of combinatorial proofs for this statement. For instance,
Quotient of factorials
However, I am wondering whether there is a purely number-theoretic proof.
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But why does any $\Gamma(z)\Gamma(1-z)\gets\pi\csc(\pi z)$?
So I was looking through my questions for a sense of nostalgia when I came across this question of mine asking on how to evaluate $$\int_{-\infty}^\infty\Gamma(1+ix)\Gamma(1-ix)dx$$Now here's the ...
- 1,661
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Integral of $\int^{2\pi}_0 \sin^{2n}(x) \cos^{2m}(x)\ dx$
I have tried to solve the following integral:
$$\int^{2\pi}_0 \sin^{2n}(x) \cos^{2m}(x)\ dx$$
And obtained the following using reduction:
$$2\pi\sum_{k=0}^{m}\left(-1\right)^{m-k}\cdot\binom{m}{k}\...
1
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1
answer
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Combinatorics question - Round of 8 draw
"In the Champions League Round of 8 draw on some year, the clubs present are: Barcelona, Bayern, Benfica, Inter Milan, Liverpool, Man City, PSG and Real Madrid.
a) Of all the draws possible, in ...
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Question about the divisibility of a sum
In this post , the function $$f(n):=\sum_{j=1}^n j!^2$$ is mentioned. $f(n)$ seems to be squarefree for every positive integer $n$.
Do we have $n+1\mid f(n)$ for some positive integer $n$ ? The ...
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1
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Show that $\sum_{m=1}^nm\cdot m!=(n+1)!-1 \forall n\ge 1$ [duplicate]
I want to show that
$\sum_{m=1}^nm\cdot m!=(n+1)!-1 \forall n\ge 1$, however I am not sure how to do that.
Let $m!=m(m-1)!$ so we replace it in for $m!$ and get
$\sum_{m=1}^nm\cdot m(m-1)! $
which ...
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A very interesting integral: $\int\limits^{2}_{1}\sqrt{x-\sqrt{x!-\sqrt{(x!)!-\sqrt{((x!)!)!-\dots}}}}dx$
I really like infinitely nested radicals so when I was messing around with them in Desmos, I noticed that the domain of $\sqrt{x-\sqrt{x!-\sqrt{(x!)!-\sqrt{((x!)!)!-\dots}}}}$ seemed to converge to $1&...
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Limit as $n$ tends to infinity of indeterminate form $0 \cdot \infty$ with factorials [closed]
How can I calculate the following limit properly?
$$\lim_{n\to\infty} (5n)^{1/5}\cdot\left(\frac{(2n)!}{(3n)!}\right)^{\frac{1}{5n}}$$
I know that the first approaches infinity while the second ...
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Does using pre-computed squares speed up significantly the calculation of factorial $n!$?
There are many different methods that tried to improve the calculation of $n!$. Few of them managed to halve the number of mulitplications. One of those methods is the basis to completely remove the ...
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Why does a sum of factorials behave differently from single factorials?
The Brocard Problem shows three factorials that can be expressed as $n!+1=m^2$ or equivalently $n!=k(k+2)$. So any time a single factorial can be put in the form of $k(k+2)$, it will belong to Brocard ...
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1
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How long before someone is ahead by 10 in a game of flipping a coin?
If me and a friend are playing a game and start with 10 marbles each. We flip a coin each round and each time it's a head he gives me a marble and each time it's tails I give him one. How many coin ...
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1
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Binomial proof of $n! = n^r - n(n - 1)^r + \frac{n(n-1)}{2!}\left(n -2 \right)^r - \dots$ when $r = n$
I realize proofs for this identity have been asked before, but the ones I've found so far involve either induction, generating functions, counting number of functions, etc. I've yet to find this ...
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Solve for f(x) if f(x)f(-x)=g(x) [closed]
Solve for $f(x)$ if $f(x)\cdot f(-x)=g(x)$.
I have been having trouble figuring this out. I asked ChatGPT, and its answers don't work. I then looked online (I googled it), and was surprised to find ...
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