Questions on the factorial function, $n!=n\times(n-1)\times\cdots\times1$. Consider using the tag (gamma-function) if dealing with noninteger arguments.

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5
votes
2answers
81 views

Challenge: How to prove this identity between bi- and trinomial coefficients?

This question is the continuation of its predecessor. Using the convention that trinomial coefficients $$ \binom{n}{k_1,k_2,k_3}=\frac{n!}{k_1! k_2! k_3!} $$ are zero if $k_i<0$ or $\sum_i k_i\neq ...
2
votes
0answers
65 views

Checking whether it is integer or not.

I'm trying to prove following term is integer for all $m,n \in \mathbb{N} :$ $$\frac{2^{m+n-1}\prod\limits_{k=1}^m(4k-3)\prod\limits_{l=1}^n(4l-1)}{(m+n)!}$$ I checked that if $m=n$ then this term ...
0
votes
2answers
79 views

Prove that $6! \mid n(n+1)…(n+5)$ [closed]

Prove that for all $n \in \mathbb{Z}$, $6! \mid n\cdot(n+1)\cdots(n+5)$ using only criteria of divisibility (without using combinatorial arguments).
3
votes
0answers
69 views

If $x$ and $y$ are positive integers then $\frac{(2x)!(2y)!}{x!y!(x+y)!}$ is an integer

If $x$ and $y$ are positive integers then $\frac{(2x)!(2y)!}{x!y!(x+y)!}$ is an integer I have to show that the proposition above is true for any $x,y\in\mathbb{Z^+}$ by means of Legendre's ...
2
votes
2answers
44 views

Infinite sum of ratio of factorials

Mathematica tells me that for $r, p$ positive integers with $\mathcal{r} < p$ we have $$\sum_{i=p+1}^\infty \frac{(i-r)!}{(i+r)!} = \frac{(p+1-r)!}{(2r-1)(p+r)!}. $$ Can anyone point me in the ...
3
votes
1answer
71 views

Proving that if $n>2$ then $n!>n^{n/2}$ using induction. [duplicate]

How to prove that if $n>2$ then $n!>n^{n/2}$ using induction?
3
votes
2answers
100 views

Determining whether a number is a 'factorial number'

Let's define $\mathbb{F}=\{x \in \mathbb{Z^+}: x=n!, n=\mathbb{N}\}$ to be the set of all 'factorial numbers' (i.e. all positive integers which are the factorial of some natural number). Is there ...
1
vote
2answers
53 views

How to reduce large combinations?

The result of a hypergeometric distribution question that I posted about earlier this evening is what follows: $$\frac{{30 \choose 10}{20 \choose 5}}{{50 \choose 15}}$$ This becomes: ...
0
votes
2answers
24 views

Factorial Summation Problem [duplicate]

$$\sum_{j=0}^n j\cdot j!$$ I got $(n+1)!-1$ as the answer but I'm not sure if that's right or how I even got to that answer exactly. (my paper is a mess of random work and I can't make it out). Can ...
1
vote
1answer
51 views

Complexity of computing $N!$

Question: Complexity of computing $N!$, considering that each multiplication cost about $O(\log^2{n})$. Attempt: There's $n-1$ multiplication. Each multiplication leads to a bigger number, thus $n-1$ ...
2
votes
2answers
59 views

Factorial as a sum. Insight appreciated

I recently posted an answer to a question about ways to express the factorial function as a sum. I posted the following formula, which I discovered several years ago and I haven't seen anywhere else: ...
1
vote
2answers
26 views

Represent non-integer values on the factorial base

I want to compute the representation of the following values using the factorial number system: $\pi$ $e$ $\phi$ I know how to do it for integer values, but is it feasible for non-integer values? ...
6
votes
1answer
193 views

Finding every triplet $(n,a,b)$ such that $n!=2^a-2^b$

Question : Let $n,a,b$ be positive integers. Are there infinitely many triplets $(n,a,b)$ which satisfy the following equality?$$n!=2^a-2^b$$ If Yes, then how can we prove that? If No, then how ...
2
votes
1answer
87 views

Challenge: How to prove this reduction identity for factorials of even numbers?

Some time ago, as a by-product of a proof, I came across an odd (at least to me) identity for reducing the factorial of an even number into a sum: $$(2n)!=\sum_{k=0}^{\lfloor \frac{n}2 \rfloor} ...
2
votes
3answers
96 views

Another question about ratios of Pochhammer symbols

My question is similar to this question. Can $$\frac{(11/6)_n (7/6)_n (3/2)_n}{(3)_n}$$ be expressed 'nicely' in terms of factorials just like $(1/6)_n (1/2)_n (5/6)_n$ in the aforementioned question? ...
0
votes
2answers
35 views

Factorial Equivalencies Calculus

Why is: (2n+1)! = (2n)(2n+1)(2n-1)! Using this, I can deduce that: (n+1)! = (n)(n+1)(n-1)! I am working with Calculus ...
1
vote
2answers
49 views

Using induction to prove that $\sum_{r=1}^n r\cdot r! =(n+1)! -1$

Use induction to prove that $\displaystyle\sum_{r=1}^n r\cdot r! =(n+1)! -1$ I first showed that the formula holds true for $n=1$. Then I put n as $k$ and got an expression for the sum in ...
8
votes
8answers
174 views

Telescoping series of form $\sum (n+1)\cdot…\cdot(n+k)$

Wolfram Alpha is able to telescope sums of the form $\sum (n+1)\cdot...\cdot(n+k)$ e.g. $(1\cdot2\cdot3) + (2\cdot3\cdot4) + (...) + n(n+1)(n+2)$ How does it do it? EDIT: We can rewrite as: $\sum ...
0
votes
1answer
16 views

Coefficients of a polynomial representation of factorials

I'm trying to figure out the coefficients of $${(k+d)!}/{(k-n)!}$$ when expressed as a polynomial. Any ideas?
2
votes
3answers
122 views

Proving that $(xy)!/y!^x$ is an integer

I'm learning about factorials and combinatorics in class, and this problem came up, but I don't know how to solve it. The teacher said that it would be an integer, but how can I show this? $$ ...
6
votes
4answers
186 views

Closed-forms of infinite series with factorial in the denominator

How to evaluate the closed-forms of series \begin{equation} 1)\,\, \sum_{n=0}^\infty\frac{1}{(3n)!}\qquad\left|\qquad2)\,\, \sum_{n=0}^\infty\frac{1}{(3n+1)!}\qquad\right|\qquad3)\,\, ...
3
votes
1answer
61 views

Any nice way to find number number of single digit ordered pairs $(a, b)$ such that $a!b! \gt a!+b!$

I have listed them all by brute force : a = 0,1 : no solutions a = 2 : b = 3,4,5,...9 c = 3 : b = 2,3,4...9 I'm wondering if there is a clever approach to ...
4
votes
4answers
200 views

Finding $\frac{\mathrm d}{\mathrm dx} x!$

I'm trying to differentiate $x!$ but I just can't seem to do it right. I define the function as follows: $$x! = \prod_{r = 0}^{x}(x-r) \quad,\quad x \in \mathbb N$$ I've tried attempted to try it by ...
1
vote
4answers
53 views

How to find the remainder when the following series is divided by 12? [duplicate]

$1! + 2! + 3!+\cdots + 99! + 100!$ I am not getting any idea on how to solve this problem. I know that modular arithmetic should be used but not getting how to start off with the solution. Please ...
1
vote
0answers
30 views

Counting zeros in a factorial(terminal + zeros in between digits)

The usual counting zeros in a factorial asks to count only the terminal zeros.This question, which also asks to count the zeros that are in between digits,for example, 8! (40320, has a zero between 4 ...
1
vote
1answer
65 views

Prove $1! + 2! + . . . + n! < (n + 1)!$ using mathematical induction [duplicate]

$1! + 2! + . . . + n! < (n + 1)!$ This question has left me stumped for quite some time. I am not sure how to approach it. (I am really bad at induction).
2
votes
1answer
114 views

Use induction to prove the following: $1! + 2! + … + n! \le (n + 1)!$

Use induction to prove the following: $1! + 2! + .... + n! < (n + 1)!$ Base case: $n = 1$ $1! < 2!$ true Inductive step: Assume that $1! + 2! + .... + k! \le (k + 1)!$ is true let $n = k ...
16
votes
3answers
274 views

Approximating $100!$

I participated in an Estimathon (a speed contest of Fermi problems) not long ago. It works as follows. Contestants are given questions and they must give a closed range $[a,b]$ which should contain ...
2
votes
2answers
69 views

How to find the limit of $ {n! e^n}/{n^{n+1/2}} $?

What is the value of this limit and how to find it? $$ \lim_{n \to \infty} \frac{n! e^n}{n^{n+\frac{1}{2}}} $$ Can we use L'Hospital rule here? I tried but failed that how to do it.
2
votes
1answer
82 views

Proving combinatorial identity with the product of Stirling numbers of the first and second kinds

$$ \sum_{k} \left[\begin{array}{c} n\\k \end{array}\right] \left\{\begin{array}{c} k\\m \end{array}\right\} = {n \choose m} \frac{\left( n-1\right)!}{\left(m-1 \right)!}, \quad \text{for } n,m > 0 ...
0
votes
1answer
48 views

Fundamental principale of permutations

I have just begin to learn about Permutation and combination. (Just learned definition and factorial.) In which i have learn: $\sideset{_n}{_r}P=n(n-1)(n+1) \dots (n-r+1)$, $r\le n$ where ...
4
votes
1answer
105 views

What is the mathamatical term for this programming concept?

In python's itertools, there is a function called permutations. It returns the number of ways to arrange x number of variables into a given space. For example, ...
-1
votes
1answer
49 views

Simplify factorials into a combinatorial formula

Is there any way to simplify this into a combinatorial formula? $$\frac{t!(n-t)!}{n!}$$
8
votes
1answer
93 views

Integer solutions of the factorial equation $(x!+1)(y!+1)=(x+y)!$

The problem is: are there solutions for the next equation? $$(x!+1)(y!+1)=(x+y)!$$ with $x,y\in\mathbb{N}$. My solution: $\left(x!+1\right)\cdot \left(y!+1\right) = \left(x+y\right)!$ ...
3
votes
1answer
117 views

What is Factorial of Zero Cubed?

My brother brought something to my attention earlier this morning and I cannot find the answer with just a googling to end the argument, so I have come to you to ask and understand. (0! 0! 0!) = n ...
2
votes
2answers
70 views

Why does this sum equal to (4^n -1)

How do I get to this solution? $\sum _{k=1}^n\left(\binom nk 3^{n-k}\right)=\left(4^n-1\right)$ I believe it's connected to this, which I know is true: $\sum \:_{k=1}^n\binom nk=2^n-1$
2
votes
2answers
75 views

Prove the inequality $n!\lt n^{n+\frac12} e^{-n+1}$ [closed]

Prove the following inequality: $$n!\lt n^{n+\frac12} e^{-n+1}.$$ Try to avoid induction if possible. Thanks!!
1
vote
0answers
124 views

The closed form of $\sum_{n=1}^{x}n!$

Let $$y=\sum_{n=1}^{x}n!$$ be the sum of consecutive factorials. What is closed form for $y$ in terms of $x$? Wolfram Alpha says that $$y=-(-1)^x\Gamma(x+2)(!(-x-2))-!(-1)-1$$ where $!x$ is ...
9
votes
1answer
299 views

Complex Factorial Equaling One

For what complex values of $z$ is $$z! =1? $$ Are they even all known? Are there finitely many or infinitely many? (Yes, the trivial $z$ are 0 and 1. )
2
votes
1answer
56 views

Given $n$, find smallest number $m$ such that $m!$ ends with $n$ zeros

I got this question as a programming exercise. I first thought it was rather trivial, and that $m = 5n$ because the number of trailing zeroes are given by the number of factors of 5 in $m!$ (and ...
1
vote
3answers
61 views

Prove that if $n\in\mathbb{N}$ and $n \geq 3$ then $n! + 3$ is composite.

Prove that if $n\in\mathbb{N}$ and $n \geq 3$ then $n! + 3$ is composite. I tried factoring it to show that there are two factors, thus composites but I can't figure out how to get rid of the ...
5
votes
1answer
77 views

Expressing a recursively defined function in terms of factorials or gamma function

Given the recursion $$f(n) = nf(n-1) + (n-1)f(n-2) $$ $$f(0) = 1, f(1) = 1$$ How exactly does one express the target function? I know that $$f(n) = nf(n-1)$$ gives rise to $$f(n) = ...
7
votes
4answers
170 views

A closed form of $\sum_{k=0}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right)$

I am looking for a closed form of the following series \begin{equation} \mathcal{I}=\sum_{k=1}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right) \end{equation} I have no idea how to ...
2
votes
0answers
61 views

Factorial of Complex Values

Since the gamma function is an analytic continuation of the factorial function, we can find the factorial of complex values. How does one go about doing so? I've looked far and wide on the internet ...
0
votes
3answers
84 views

Multiplication of the numbers $1$ to $n$

Let's say I want to find the product of $1,2,3, \dots, 10$. Do I need to do $1 \cdot 2 \cdot 3 \cdot \dots \cdot 10$ manually or is there an easier way to do it? Something like the sumation of $1$ ...
0
votes
0answers
28 views

number of trailing zeroes of factorial raise to power by another factorial

Finding trailing zeroes in any factorial is easy. Every time you pass a multiple of 10 (or something 5 mod 10) you will accumulate another 0 For example 10! has two trailing zeros, one from ...
1
vote
1answer
49 views

Limit of factorial how to continue

$$\lim_{n\to\infty}\left(\dfrac{(n+1)^{n+1}\cdot n!}{2^{(n+1)!-n!}}\right)=\lim_{n\to\infty}\left(\dfrac{(n+1)^{n+1}\cdot n!}{2^{n!\cdot n}}\right).$$ How to continue? the answer is $0$ ... thank you ...
4
votes
2answers
220 views

N! ends with exactly 30 zeros? [duplicate]

How many values of N exist, such that N! ends with exactly 30 zeros?
0
votes
1answer
50 views

Factorial Taxicab Number

What is the $i$th number $T_! (n,k,i)$ such that $T_! (n,k,i)$ is the sum of $n \in \mathbb{N}$ distinct positive integer factorials in $k \in \mathbb{N}$ distinct ways (ignoring ordering, ...
6
votes
3answers
123 views

The relation between the number of $0$s which are at the end of $3^{n!}-1$ and that of $n!$

Let $a_n,b_n$ be the number of $0$s which are at the end of $3^{n!}-1,n!$ in the decimal system respectively. I found that $a_n=b_n+1$ holds for $n=4,5,\cdots, 10$. Then, my questions are... ...