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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Semi-Factorial Skipping Every $k^\text{th}$ Number

For an integer $n$, the semi-factorial $n!!$ can be defined as $$ n!! = n(n-2)(n-4)\cdots $$ In other words, the semi-factorial of $n$ is the familiar factorial, but with every other term omitted. For ...
3
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3answers
496 views

Infinite Series using Falling Factorials

I recently started reading Concrete Mathematics by Graham, Knuth and Patashnik and met falling/rising factorials for the first time; it seemed like a very convenient method for evaluating particular ...
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2answers
181 views

How do I show the following property of a factorial?

How do I show the following? $$\frac{n!}{(k+1)!(n-(k+1))!}=\frac{n-k}{k+1}\frac{n!}{k!(n-k)!} \text{ for } k=0,1,\ldots,n-1$$
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7answers
203 views

Some trouble with a proof on $n!/(\sqrt{n})^n \geq 1$

Originally the problem is to prove that $n! \geq n^{n/2}$. I reduced this to: $n! \geq (\sqrt{n})^n$ so that: Prove that $\frac{n!}{(\sqrt{n})^n} \geq 1$. Each term in $n!$ is divided by the ...
19
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1answer
481 views

Integer solutions of $x! = y! + z!$

There was an interesting problem asked about triples $(x,y,z)$ which are solutions of $$x! = y! + z!.$$ Here $(2,1,1)$ is a solution because $2! = 1! + 1!$, as are $(2,1,0)$ and $(2,0,1)$. Now I ...
2
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1answer
2k views

Opposite of Factorial? [duplicate]

Possible Duplicate: Is there a way to solve for an unknown in a factorial? I was just wondering, what would be the opposite of factorial? For example, If I had $n! = 120$. How can I then ...
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4answers
962 views

Simplify a factorial

I have the problem to evaluate the following: $$ (2n)!\over 2^n(n!) $$ Does this reduce to anything in particular? I stuck it into a computer and it's ...
2
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1answer
224 views

Proof the following proposition: for all $n \geq 0, \mathrm{fib}(n) \leq n!$

I am a comp science undergrad and just started to learn proof. And I have been thinking about this question for a few days. How should I present my answer? Do I have to use the Binet's formula? Or can ...
2
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2answers
469 views

How many bits are in factorial?

I am interested in good integer approximation from below and from above for binary Log(N!). The question and the question provides only a general idea but not exact values. In other words I need ...
49
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3answers
2k views

Why is $i! = 0.498015668 - 0.154949828i$?

While moving my laptop the other day, I ended up mashing the keyboard a little, and by pure chance managed to do a google search for i!. Curiously, Google's ...
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4answers
2k views

Showing that $\frac{\sqrt[n]{n!}}{n}$ $\rightarrow \frac{1}{e}$

Show:$$\lim_{n\to\infty}\frac{\sqrt[n]{n!}}{n}= \frac{1}{e}$$ So I can expand the numerator by geometric mean. Letting $C_{n}=\left(\ln(a_{1})+...+\ln(a_{n})\right)/n$. Let the numerator be called ...
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2answers
288 views

How to find a large factorial in scientific notation?

For example, given $8952!$, how do I write this in scientific notation?
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0answers
142 views

Generating all positive integers from three operations

This question arose on sci.math, but since almost all the competent mathematicians there have migrated here, I thought I'd give the question a wider audience. Starting with an integer $t>2$, ...
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3answers
120 views

Closed form for $T(1) = K, T(x) = xT(x-1) + x$?

I'm looking for a closed form for the following recurrence: $T(1) = K$ $T(x) = xT(x-1) + x$ I know it is factorial-like but I am unable to get an exact answer.
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3answers
257 views

Maple Error on Asymptotic Analysis of $\ln(n)!$

In Maple, the command asmypt($f$,$x$) computes the asymptotic expansion of the function $f$ with respect to the variable $x$ (as $x \rightarrow \infty$). The command asympt(ln(n)!,n); gives the ...
8
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1answer
220 views

The first column of the $n$th power for a triangular matrix

I have found a interesting thing but I cannot prove it. Given $k_i$ are positive for any $i\geq1$, and we have $M+1$ by $M+1$ matrix $A$, which is $$ A=\left[\begin{array}{ccccc} 0\\ k_{1} & 0\\ ...
8
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6answers
493 views

Prove that for any $x \in \mathbb N$ such that $x<n!$ is the sum of at most $n$ distinct divisors of $n!$

Prove that for any $x \in \mathbb N$ such that $x<n!$ is the sum of at most $n$ distinct divisors of $n!$.
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0answers
136 views

Prove: $\frac{(2px)!}{((px)!)^2}\equiv\frac{(2x)!}{((x)!)^2}\pmod{p^2}$

How can I prove the following, where $p$ is a prime and $x$ a positive integer? $$\dfrac{(2px)!}{((px)!)^2}\equiv\dfrac{(2x)!}{((x)!)^2}\pmod{p^2}$$ I'm not sure if it is actually true, but I tested ...
3
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6answers
558 views

Proof by contradiction that $n!$ is not $O(2^n)$

I am having issues with this proof: Prove by contradiction that $n! \ne O(2^n)$. From what I understand, we are supposed to use a previous proof (which successfully proved that $2^n = O(n!)$) to find ...
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3answers
469 views

Constructing $\mathbb N$ from the set of factorials

Let S be the set $\{0!, 1!, 2!, \ldots\}$. Is it possible to construct any positive integer using only addition, subtraction and multiplication, and using any element in S at most once? For example: ...
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1answer
137 views

How to compare big numbers that are outcome of different functions.

How is the best way to compare big numbers? They are result of two functions with different asymptotic growth. For example: Googleplex which is $10^{{10}^{100}}$ to $1000!$
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3answers
133 views

closed-form expressions for product of 3n+k where k = 1 or 2

There are some easy products that can be written in closed form in terms of factorials: $ 2 \times 4 \times 6 \times ... 2n = n! \times 2^n$ $ 1 \times 3 \times 5 \times ... (2n-1) = {{(2n)!} ...
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3answers
3k views

Limit of a sequence involving root of a factorial: $\lim_{n \to \infty} \frac{n}{ \sqrt [n]{n!}}$ [duplicate]

I need to check if $$\lim_{n \to \infty} \frac{n}{ \sqrt [n]{n!}}$$ converges or not. Additionally, I wanted to show that the sequence is monotonically increasing in n and so limit exists. Any help is ...
4
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1answer
5k views

Is there a way to reverse factorials? [duplicate]

Possible Duplicate: Is there an Inverse Gamma $\Gamma^{-1} (z) $ function? Is there any way I can 'undo' the factorial operation? JUst like you can do squares and square roots, can you do ...
9
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4answers
295 views

Is $10^{8}!$ greater than $10^{10^9}$?

My question is: $10^8! > 10^{10^9}$ ? I know that factorial is greater than exponential, but I am not sure about this specific case. Thanks,
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3answers
223 views

Limits defined for negative factorials (i.e. $(-n)!,\space n\in\mathbb{N}$)

I apoligize if this is a stupid/obvious question, but last night I was wondering how we can compute limits for factorials of negative integers, for instance, how do we evaluate: ...
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1answer
176 views

Does a closed form formula for the series ${n \choose n-1} + {n+1 \choose n-2} + {n+2 \choose n-3} + \cdots + {2n - 1 \choose 0}$ exist.

$${n \choose n-1} + {n+1 \choose n-2} + {n+2 \choose n-3} + \cdots + {2n - 1 \choose 0}$$ For the above series, does a closed form exist?
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3answers
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Proof that $\frac{(2n)!}{2^n}$ is integer

I am trying to prove that $\dfrac{(2n)!}{2^n}$ is integer. So I have tried it by induction, I have took $n=1$, for which we would have $2/2=1$ is integer. So for $n=k$ it is true, so now comes time ...
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3answers
250 views

How to approximate $\sum_{k=1}^n k!$ using Stirling's formula?

How to find summation of the first $n$ factorials, $$1! + 2! + \cdots + n!$$ I know there's no direct formula, but how can it be estimated using Stirling's formula? Another question : Why can't ...
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2answers
1k views

Given number of trailing zeros in n!, find out the possible values of n.

It's quite straightforward to find out number of trailing zeros in n!. But what if the reverse question is asked? n! has 13 trailing zeros, what are the possible values of n ? How should we ...
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3answers
317 views

Summation of a factorial (total number terms in a polynomial)

By induction I can prove : $$\sum^{M}_{t=0}\frac{(t+D-1)!}{t!(D-1)!} = \frac{(D+M)!}{D!M!} $$ However, I couldn't derive the right hand side directly. It would be of great help if anyone can solve ...
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3answers
116 views

Formula to calculate the number of possible positions for $x$ numbers

What formula do I use to calculate the number of possible positions for $x$ numbers? Let's say I have $3$ people in a race. What are all the possible combinations of the order they can finish in? ...
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6answers
1k views

$m!n! < (m+n)!$ Proof?

Prove that if $m$ and $n$ are positive integers then $m!n! < (m+n)!$ Given hint: $m!= 1\times 2\times 3\times\cdots\times m$ and $1<m+1, 2<m+2, \ldots , n<m+n$ It looks simple but ...
2
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1answer
53 views

$t > 2n^2 \implies t!>n^t$ for $n,t \in \mathbb{N}$

I have come across this in a proof: If $t>2n^2$ then, $$t!>(n^2)^{t-n^2}=n^tn^{t-2n^2}>n^t$$ Obviously, this is much help to determine the relationship between factorials and exponential, ...
0
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1answer
148 views

The mathematics underlying this baroque expression of the double-factorial

On the OEIS page for the double factorial, there are three ways of getting the sequence in PARI. One of them is this curious bit of PARI code: ...
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2answers
455 views

Find the remainder when $ 12!^{14!} +1 $ is divided by $13$

Find the remainder when $ 12!^{14!} +1 $ is divided by $13$ I faced this problem in one of my recent exam. It is reminiscent of Wilson's theorem. So, I was convinced that $12! \equiv -1 \pmod {13} $ ...
7
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1answer
732 views

How to prove that $\mathrm{Fibonacci}(n) \leq n!$, for $n\geq 0$

I am trying to prove it by induction, but I'm stuck $$\mathrm{fib}(0) = 0 < 0! = 1;$$ $$\mathrm{fib}(1) = 1 = 1! = 1;$$ Base case n = 2, $$\mathrm{fib}(2) = 1 < 2! = 2;$$ Inductive case ...
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0answers
66 views

Is there a standard name or shorthand for “plustorial”? [duplicate]

Possible Duplicate: What is the term for a factorial type operation, but with summation instead of products? We're all familiar with factorial: $$n>0,\quad n! = n \times (n-1) \times ...
7
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1answer
502 views

Ramanujan's approximation to factorial

I saw this approximation for the factorial given by Ramanujan as $$\log(n!) \approx n \log n - n + \frac{\log(n(1+4n(1+2n)))}{6} + \frac{\log(\pi)}{2}$$ in wikipedia, which claims the approximation is ...
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2answers
785 views

factorial Vs power sequence

What sequence is dominant between $ f(n) = n!$ and $ g(n) = 2^n$ or (or $a^n$) I mean $ f/g -> 0 $ or $infinity$
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0answers
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How to find the last non-zero digit in ${^n\!P_k} $?

What is the procedure of finding the last non-zero element in ${^n\!P_k}$?
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1answer
282 views

Solutions of $p!q! = r!$

The title says it all, more or less. Obviously, there are infinitely many "trivial" integral solutions of the form $p=n, q=(n!-1), r= n!$. How many non-trivial solutions are there? I came across this ...
2
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1answer
182 views

If a function is a product of factorials of polynomials, how can I prove that this representation is unique?

If a function is a product of factorials of polynomials, how can I prove that this representation is unique? Specifically, let $F(x) = \prod_i{P_i(x)!^{a_i}}$, $F:\mathbb{N} \rightarrow \mathbb{Q}$, ...
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2answers
217 views

Relating Gamma and factorial function for non-integer values.

We have $$\Gamma(n+1)=n!,\ \ \ \ \ \Gamma(n+2)=(n+1)!$$ for integers, so if $\Delta$ is some real value with $$0<\Delta<1,$$ then $$n!\ <\ \Gamma(n+1+\Delta)\ <\ (n+1)!,$$ because ...
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2answers
152 views

Upper bound for the series $\sum_{n\geq 1}\frac{1}{(n+1)^{a+1}}\sum_{k=0}^n b^k\left(\frac{(n-k)!}{n!}\right)^a$

I want to show that the series $$\sum_{n\geq 1}\frac{1}{(n+1)^{a+1}}\sum_{k=0}^n b^k\left(\frac{(n-k)!}{n!}\right)^a$$ converges for $a,b>0$. I have tried this so much that the smallest hint will ...
2
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1answer
42 views

Limit of $\frac{a^{n+1}(n+1)!^b}{\sum_{k=0}^n a^kk!^b}$ when $n\rightarrow\infty$

I want to prove that $$\lim_{n\rightarrow\infty}\frac{a^{n+1}(n+1)!^b}{\sum_{k=0}^n a^kk!^b}<\infty$$ for $a,b>0$. This is the last step of a bigger problem. I believe it would suffice to use ...
23
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2answers
3k views

How many zeroes are in 100!

One common math puzzle I've seen around asks for how many zeros are in the product of "100!" Usually, the solution everyone gives goes something like try to match pairs of 5s and 2s that factor out ...
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4answers
3k views

Highest power of a prime $p$ dividing $N!$

How does one find the highest power of a prime $p$ that divides $N!$ and other related products? Related question: How many zeros are there at the end of $N!$? This is being done to reduce ...
2
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2answers
2k views

Why is $\log(n!)$ $O(n\log n)$?

I thought that $\log(n!)$ would be $\Omega(n \log n )$, but I read somewhere that $\log(n!) = O(n\log n)$. Why?
5
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3answers
508 views

What is the definition of $2.5!$? (2.5 factorial)

I was messing around with my TI-84 Plus Silver Edition calculator and discovered that it will actually give me values when taking the factorial of any number $n/2$ where $n$ is any integer greater ...