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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1answer
59 views

Proving a sequence with induction reasoning

I have an assignment which I am quite stuck on. The question is the following: function f: N to N is defined recursivly: ...
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1answer
90 views

What's a useful recurrence relation for $(2n)!$ in terms of $n!$?

I want to make an algorithm for $n!$ which will divide the number in half and call the algorithm again until n is so close to $0$ that a value of $1$ can be safely returned, and use the value of each ...
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1answer
60 views

Limits and common sense

I'm stuck in understanding of limits. It all makes sense, but at a certain point my answers which seem logical to me are not true. Please can somebody explain why as a huge number gets divided by a ...
1
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1answer
63 views

Limit of a function not using Stirling's Approximation

I want to compute the following limit: $$\lim_{n\to\infty} \frac{\left(\frac{e}{F_{n+1}}\right)^{F_{n+1}} F_{n+1}!}{\left(\frac{e}{F_n}\right)^{F_n} F_n!},$$ where $F_n$ is the $n$th Fibonacci ...
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1answer
71 views

How to derive $\frac{d}{dn}n! = \Gamma(1 + n)\psi^{(0)}(n+ 1)$?

How can you derive $\frac{d}{dn}n! = \Gamma(1 + n)\psi^{(0)}(n + 1)$? I have tried checking Wolfram Alpha for a step-by-step solution, but none is given. Moreover, of what is the second function, ...
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1answer
49 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, ...
0
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1answer
55 views

Simpler expression for $\sum_{k=1}^{n}{k!}$

Is there a way to express $$\sum_{k=1}^{n}{k!}$$ in a simpler way that doesn't use sums up to n ? I've searched for this around the web and found that the subfactorial function can help with this, ...
0
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1answer
47 views

help on manipulating this algebraic expression

So I have something like: $\frac {k!}{(k-3)!3!}$ I'm going to add $\frac 12k(k-1)$ to this, and I want to obtain $\frac {(k+1)!}{(k-2)!3!}$ as the result. I'm having trouble with this since I need ...
0
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1answer
87 views

Theory behind multiplication & combinations?

If with the Binomial Coefficient we try to find the possible combinations $\binom{n}{k}$ where $n$ is equal to $k$ what is the theory behind factorial resulting in the correct solution? E.g. ...
0
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1answer
32 views

Simplify the following problem

How $$\frac{1}{k}\sum_{m=r}^{k-1}\frac{m!}{(m-r)!}=\frac{(k-1)(k-2)\ldots(k-r)}{r+1};\quad r=1,2,\ldots$$ I have thought in two ways: ...
8
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0answers
427 views

Understanding Ramanujan's approach in his proof of Bertrand's Postulate

I've been reading through Ramanujan's proof of Betrand's Postulate and I'm not clear why he didn't state his proof in terms of $\varphi(2x) - \varphi(x)$ What would be wrong with this approach for ...
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0answers
2k views

When does a Square Matrix have an LU Decomposition?

When can we split a square matrix (rows = columns) into it’s LU decomposition? The LUP (LU Decomposition with pivoting) always exists; however, a true LU decomposition does not always exist. How do ...
4
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0answers
64 views

Trying to show that $\ln(x!) - \ln(\lfloor\frac{x}{2}\rfloor!) - \ln(\lfloor\frac{x}{3}\rfloor!) - \ln(\lfloor\frac{x}{6}\rfloor!) \ge \psi(x)$

I've been told that the approach below will not work. I would be interested if someone could help me to understand what will go wrong. Let: $$\psi(x) = \sum\limits_{p^k \le x} \ln p$$ So that (see ...
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0answers
94 views

Generalizing Jitsuro Nagura's result: my resulting upper bound for the second chebyshev function is too low. What am I doing wrong?

I've been reading through Jitsuro Nagura's classic proof that there is a prime between $x$ and $\frac{6x}{5}$ and it seems to me that it should be possible to improve on his upper bound for the second ...
4
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0answers
136 views

Inequality problem with factorials

I am not sure if this kind of "question" is welcome on MSE. Here is an olympiad-like problem that I would like to share with you: Let $a,b,c$ be nonnegative integers. Prove that $$ ...
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0answers
58 views

Which series of numbers effectively translates the factorial to the exponential function?

We have the relation of the Bernoulli numbers $$B_{2n} = (-1)^{n+1}\frac {2(2n)!} {(2\pi)^{2n}} \left(1+\frac{1}{2^{2n}}+\frac{1}{3^{2n}}+\cdots\;\right).$$ For $n>1$, the right hand sum ...
3
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0answers
96 views

Prove (*) by induction on k.

Challenge: For linear systems with constant coefficients, in some sense we "never need more" than the so-called exponential polynomials, meaning expressions in the form $$\sum_{i=1}^m ...
3
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0answers
133 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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0answers
194 views

How to find $\beta$ and $\alpha$?

$\mathbb{P}$ is the prime numbers set. $p \in \mathbb{P}$ $a,b,c \in \mathbb{N}$ $n=a p^b+c$ where $c= n\bmod p$ $b$ is the highest power of $p$ who divides $n-c$ How to find $\beta$ where ...
2
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0answers
51 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 ...
2
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0answers
32 views

Knuth shuffle : Is there a reciprocal to the factorial?

I have looked into the Knuth collection shuffle algorithm with pseudorandom number generators. They say that a PRNG with a seed state of $19937$ bits (like one of the Mersenne Twisters) can shuffle a ...
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0answers
40 views

Find k such that, $\displaystyle\frac{n^{\underline k}}{n^k}<a$

How to find k here $\displaystyle\frac{n^{\underline k}}{n^k}<a$ ? With, $n^{\underline k}=n\cdot(n-1)\cdot...(n-k+1)$ and $(n>k,\ a>0)$ Of course if $k\approx n/2$ the inequality ...
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0answers
59 views

Transforming a Riemann-Stieltjes integral related to the factorial

I have been able to show that $$\log n! = \int_{1 + \epsilon}^n \log x \, d\lfloor x \rfloor$$ but I have not been successful trying to transform this Riemann-Stieltjes integral to an ordinary ...
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0answers
70 views

Find all possible solutions!

Find solutions for $$^nP_r=s!$$ For $(n,r,s)\in \mathbb{N}$ I could find some trivial solutions $(6,3,5)~,~(1,1,1)$ etc.
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0answers
95 views

Compute $(-1)^n\sum_{k=1}^n (-1)^k\frac{(k+n-1)!}{(k-1)!(k-1)!(n-k)!}$

Compute $(-1)^n\sum_{k=1}^n (-1)^k\frac{(k+n-1)!}{(k-1)!(k-1)!(n-k)!}$ Define $a_{k,m}=\frac{(-1)^{k+m}(n+k-1)!(n+m-1)!}{(k+m-1)[(k-1)!(m-1)!]^2(n-m)!(n-k)!}$ Compute ...
2
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0answers
60 views

Trying to generalize an inequality from Jitsuro Nagura: Does this work?

I am investigating the generality of Lemma 1 in Nagura's proof that there is always a prime between $x$ and $\frac{6x}{5}$. In Lemma 1, Nagura establishes when $n > 1$, $x \ge 1$: ...
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0answers
110 views

Can we simplify the sum $\sum_{i=0}^k \frac{((i-k)a)^i b^{k-i}}{i!}$?

The problome is rewriten here: $\sum_{i=0}^k \frac{((i-k)a)^i b^{k-i}}{i!}$ where $0<a<1$, k is an integer larger than 1. I came to this equation when i try to find some probability. I ...
2
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0answers
141 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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0answers
111 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 ...
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0answers
9 views

Simplifcation of the Function Required

I want to simplify the following expression $$ \sum_{x=0}^{D+m} \frac {x!} {(n+1)_x} (\frac {\theta_{eff}} {\mu})^x {D+m \choose x} $$ The parameters $D, m, n, \theta_{eff}, \mu, $ are constants. ...
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0answers
44 views

The Number of The 0's in a Factorial

I need to find that the number of the 0's at the end of the number is odd or even in a factorial. For example: $0! = 1$ (Even) $5! = 120 $ (Odd) $18! = 6402373705728000 $ (Odd) Dou you have any ...
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0answers
63 views

Digits of $n$ factorial

With the notable exception of $0$, for large enough $n$, the digits in base $10$ for $n!$ seem pretty much uniformly distributed (I have also checked for other few bases $> 2$). Have anyone ...
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0answers
40 views

Approximating the probability that a range bounds a given number, with very large numbers

Let $m$ numbers be chosen uniformly from $0,\dots,n-1$ without replacement and then sorted in ascending order as $\ell_0,\dots,\ell_{m-1}$. Let there be $b,e,x$ such that $0 \le b \le e \le m$ and $0 ...
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0answers
58 views

Double summation of elementary functions

I am finding some trouble on calculating the following double summation: $ \sum_{k=1}^\infty \frac{b^k}{k!k}\sum_{n=0}^{k-1}\frac{(b*x)^n}{(n-1)!} $ Note that the inside sum gives: ...
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0answers
41 views

What are the best and most elementary bounds for $n!$?

What this question is looking for is bounds on $n!$ that are elementary in nature (I seem to have a fetish for these type of proofs). In general, as the results become more complicated, they also ...
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0answers
65 views

Simplify $\frac{[m+n-1]!}{[m]![n]!}$ where $[k]=x^k-x^{-k}$ and $[k]!=[2][3]…[k]$.

Adopting the notation $[k] = x^k - x^{-k} $ and $[k]! = [2][3]...[k]$ (note that $[1]$ is omitted), and letting $m,n$ be two integers greater than $1$ such that $n>m$ and $gcd(m,n)=1$, would it be ...
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0answers
83 views

Simplifying expression

I am looking for a way to simplify this expression: $$ \sum_{i=0}^{n-k-1} \sum_{j=0}^{k-1} \left[ {n-k-1 \choose i} {k-1 \choose j} ((-1)^{k-1-j} - (-1)^{n-k-1-i}) \times {(n+0.5)! \over ...
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0answers
34 views

Is it possible to generalize Ramanujan's lower bound for factorials when $\{\frac{x}{b_2}\} + \{\frac{x}{b_3}\} \ge 1$?

This is the second attempt at a proof. My first attempt had a flaw in its logic. After reviewing the mistake in logic, I believe that with a revised logic, the argument can be saved. The revision ...
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0answers
138 views

Using the gamma function as an upper and lower bound to the logarithm of a factorial function.

I am trying to find an upper and lower bound for the following function: $$f(x) = \ln(\lfloor\frac{x}{b_1}\rfloor!) - \ln(\lfloor\frac{x}{b_2}\rfloor!) - \ln(\lfloor\frac{x}{b_3}\rfloor!)$$ where ...
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0answers
143 views

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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0answers
134 views

Efficient factorion search in arbitrary base

A factorion in base $N$ is a natural number equal to the sum of the factorials of its digits in base $N$. So, the decimal factorions are: $1 = 1!$ $2 = 2!$ $145 = 1! + 4! + 5!$ $40585 = 4! + 0! + 5! ...
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0answers
31 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: $nP_r=n(n-1)(n+1)...(n-r+1); r\le n$ where $n=$distinct object that can ...
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0answers
22 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 ...
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31 views

Reasoning about factorials and powers of a finite set of primes

I am working on an answer to another question: How to prove $k!+(2k)!+\cdots+(nk)!$ has a prime divisor greater than $k!$ I've reduced the question to showing that the following infinite set of ...
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0answers
57 views

lower bound for factorials product

For $n$ positive integer, let $F(n) = 1! × 2! × 3! × 4! × \cdots × n!$, product of factorial(i) for $i$ in $[1\ldots n]$. Let $G(n) = \{i \in [1\ldots n],\text{ such that }n\mid F(i)\}$. It is ...
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0answers
65 views

Solving an equation with several summations and a falling factorial

Can you help me to solve quite a complicated inequation that contains summations? I'd like to solve it for b (leaving balone on ...
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0answers
81 views

Using Stirling's Formula to approximate a difference of logarithms of factorials in the same way as Jitsuro Nagura.

In Jitsuro Nagura's classic proof of a prime existing between $x$ and $\frac{6x}{5}$, he uses Stirling's formula to show that: $$T\left(x\right) - T\left(\frac{x}{2}\right) - ...
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0answers
54 views

Do these inequalities regarding the gamma function and factorials work?

I am seeing if it is possible to generalize lemma 1 in Nagura's proof that there is always a prime between $x$ and $\frac{6x}{5}$. In a previous question, I asked whether the following inequality is ...
0
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0answers
449 views

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 ...
0
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0answers
121 views

Is it true that $n> a^2\Rightarrow n!>a^n$, $n\in\mathbb{N}, a\in\mathbb{R}$?

If so, how can it be proven? (I have evaluated it up to $n=25$.) If not, does there exist a $k\in\mathbb{R}$ such as that $n> a^k\Rightarrow n!>a^n$, with $n\in\mathbb{N},a\in\mathbb{R}$? It ...