# Can the factorial function be written as a sum?

I know of the sum of the natural logarithms of the factors of n! , but would like to know if any others exist.

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There's many for example: $n!=(n!-7)+7$:) – user63181 Jul 25 '13 at 4:21
Hmm. Given that an integral is an "infinite sum", I think $\Gamma(n+1)=n!$ infinitesimally works. – 1015 Jul 25 '13 at 4:42

This one is pretty important: $$n! = \sum_{\sigma\in S_n} 1$$

Edit: As Arkamis explains, $S_n$ is the symmetric group on $n$ letters. Each $\sigma\in S_n$ is a permutation on the set $[1,2,\ldots,n]$. Since $S_n$ is a finite set, we may sum a function over it, and the sum of the constant function $f(\sigma)=1$ is just the size of the set, which is $|S_n| = n!$.

Arguably, summing a constant function is cheating. Here's one way to raise the stakes. Let $B_n$ be the set of $n\times n$ integer matrices $A$ such that every sum of a subset of entries from $A$ is in $[0,n]$. Then:

$$n!=\sum_{A\in B_n}|\det A|$$

This is the same identity in a more interesting disguise. Every $n\times n$ permutation matrix $A$ is a member of $B_n$, and $\det A = \pm 1$. On the other hand, if $A\in B_n$ is not a permutation matrix, then you can prove that $\det A = 0$.

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I can't tell if you are being sarcastic - but I don't know what that sum means, can you please explain? – zerosofthezeta Jul 25 '13 at 4:37
@Euclid It means that the order of the symmetric group $S_n$ is $n!$. $\sigma \in S_n$ means "each element $\sigma$ in the group $S_n$, of which there are $n!$. – Emily Jul 25 '13 at 4:38
Nice, haha$\left . \right .$ – Thomas Jul 25 '13 at 5:15

$n!$ is $n$ groups of $(n-1)!$ objects, so $n! = \sum_{k=1}^n (n-1)!$.

Then, $(n-1)!$ is $n-1$ groups of $(n-2)!$ objects, so $n! = \sum_{k_1=1}^n \sum_{k_2 = 1}^{n-1} (n-2)!$, and so on.

Example:

$$4! = \sum_{k_1 = 1}^4 3! = 3!+3!+3!+3! = 4\cdot 3!.$$ \begin{align*} 4! &= \sum_{k_1=1}^4 \sum_{k_2=1}^3 2! \\ &= \sum_{k_1=1}^4 2!+2!+2! \\ &= 2!+2!+2! + 2!+2!+2! + 2!+2!+2! + 2!+2!+2! \\ &= 12\cdot 2! \\ &=4\cdot 3\cdot 2!. \end{align*}

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$n! = e^{ \sum_{k = 1}^n \ln (k)}$

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Thanks, but I addressed this one in my question... – zerosofthezeta Sep 14 '13 at 7:39

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

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Sure, for $n\ge 1$ we have $$n!=\fbox{(n-1)\times (n-1)!} + \fbox{(n-1)!}$$

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If you really want, we can write $n!$ using this infinite series

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I doubt that the series converges - it is probably an asymptotic series. – marty cohen Jul 25 '13 at 4:30
The series "at $n=\infty$" is an asymptotic series. The series "at $n=0$" and the series "at $n=-1$" have radius of convergence $1$. – Robert Israel Jul 25 '13 at 4:35

$$n!=\prod_{p<n}p^{\sum_{k=1}^\infty\lfloor{n\over p^k}\rfloor}$$

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Can you provide an example for a small number like 8? – zerosofthezeta Aug 5 '13 at 3:51
This comes from prime factorization of $n!$. Any $p^e<n$ will divide $n!$. And $e=\sum_{k=1}^\infty\lfloor{n\over p^k}\rfloor.$ – Kunnysan Aug 5 '13 at 4:29
$8! = 2^{4+2+1} 3^{2} 5^{1} 7^{1}$. – Slade Dec 16 '13 at 7:34

Another combinatorially important sum: $\displaystyle n! = \sum_{k=0}^n\bigl[\begin{smallmatrix}n\\k \end{smallmatrix}\bigr]$, where $\displaystyle\bigl[\begin{smallmatrix}n\\k \end{smallmatrix}\bigr]$ is the (unsigned) Stirling symbol of the first kind, which can be defined (for instance) by the relation $x(x+1)\ldots(x+n-1) = \sum_{k=0}^n\bigl[\begin{smallmatrix}n\\k \end{smallmatrix}\bigr] x^k$. (And of course, by setting $x=1$ in this relation we get the initial sum.) The combinatorial significance is that $\bigl[\begin{smallmatrix}n\\k \end{smallmatrix}\bigr]$ counts the number of permutations of the numbers $1..n$ with $k$ distinct cycles; the relation $\displaystyle n! = \sum_{k=0}^n\bigl[\begin{smallmatrix}n\\k \end{smallmatrix}\bigr]$ thus says that the total number of permutations is just the sum over all $k$ of the number of permutations into $k$ cycles.

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I have found the following formula: $$n!=\sum_{k=2}^{n+1}(-1)^{n+1-k}\binom{n+1}{k}\sum_{i=1}^{k-1}i^{n},\ n\in N.$$

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I know this question is old, but I found it googling for summation definitions of the factorial function, and thought a good answer would be the following formula, which I discovered several years ago and I haven't seen anywhere else:

$n!=\displaystyle\sum_{k=1}^n \binom{n}{k} (-1)^{n+k} k^n$

I never took the time to find a proof, but it certainly works. It would be interesting if anyone with decent math skills (that is, better than me) could shed some light on it.

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$$\frac{(n+2)!}{2} = (n+1)! + \sum_{i=1}^n i\,n!$$

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$$(v+1)! = 1 + \sum_{k=1}^v (v-k-1)! (v-k)^2$$

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Explaining why this formula is true would be a good improvement to this answer. – apnorton Jan 27 '15 at 20:55
Sorry i have dyslexia ,change the left side of the equation with ν! – kostas Jan 28 '15 at 3:52

Here's a good reference document from a few years back: Factorials as sums by Roberto Anglani and Margherita Barile.

In this paper we give an additive representation of the factorial, which can be proven by a simple quick analytical argument. We also present some generalizations, which are linked, on the one hand to an arithmetical theorem proven by Euler (decomposition of primes as the sum of two squares), and, on the other hand, to modern combinatorics (Stirling numbers).

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While this link may answer the question, it is better to include the essential parts of the answer here and provide the link for reference. Link-only answers can become invalid if the linked page changes. – Edward Jiang Jan 28 '15 at 4:35

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