# Sums of rising factorial powers

Doodling in wolfram, I found that $$\sum^{k}_{n=1}1=k$$ The formula is pretty obvious, but then you get $$\sum^{k}_{n=1}n=\frac{k(k+1)}{2}$$ That is a very well known formula, but then it gets interesting when you calculate $$\sum^{k}_{n=1}n(n+1)=\frac{k(k+1)(k+2)}{3}\\ \sum^{k}_{n=1}n(n+1)(n+2)=\frac{k(k+1)(k+2)(k+3)}{4}$$ And so on. There is an obvious pattern that I really doubt is a coincidence, but I have no idea how to prove it in the general case. Any ideas?

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Pascal's triangle comes to mind here –  Omnomnomnom Sep 4 '13 at 3:47
@Omnomnomnom I thought about it but I was unable to introduce binomials there. However, you can generalize the hypothesis using factorials, but usually working with factorials is harder than not doing so(At least I would have a very hard time trying to do so). –  chubakueno Sep 4 '13 at 3:51
These are essentially the sums for the general $d$ dimensional simplex. –  Jaycob Coleman Sep 4 '13 at 4:02
It's overkill, but you could expand and apply en.wikipedia.org/wiki/Faulhaber%27s_formula –  dls Sep 4 '13 at 4:11
@dls I think that in that case the interesting part would go backwards: That expanding and a applying Faulhaber´s yields such a regular and simple result :) –  chubakueno Sep 4 '13 at 4:22

You can argue any given case by induction. I will take your last,$$\sum^{k}_{n=1}n(n+1)(n+2)=\frac{k(k+1)(k+2)(k+3)}{4}$$ for the example, but I think it is easy to see how it gets carried forward. The base case is simply $1\cdot 2\cdot 3=1\cdot 2\cdot 3\cdot \frac 44$ If it is true up to $k$, then $$\sum^{k+1}_{n=1}n(n+1)(n+2)\\=\sum^{k}_{n=1}n(n+1)(n+2)+(k+1)(k+2)(k+3)\\=\frac{k(k+1)(k+2)(k+3)}{4}+(k+1)(k+2)(k+3)\frac {k+4-k}4\\=\frac{(k+1)(k+2)(k+3)(k+4)}{4}$$

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Thanks! Just changing the $2$, $3$ and $4$ by $x$,$x+1$ and $x+2$ does the work of generalizing it very nicely. –  chubakueno Sep 4 '13 at 3:59

The easy way to deal, for example, with $\sum_{i=1}^n i(i+1)(i+2)(i+3)$ is to let $F(i)=i(i+1)(i+2)(i+3)(i+4)$. We calculate $F(i)-F(i-1)$. We get $$i(i+1)(i+2)(i+3)(i+4)-(i-1)(i)(i+1)(i+2)(i+3).$$ There is a common factor of $i(i+1)(i+2)(i+3)$. When we "take it out" we are left with $(i+4)-(i-1)=5$.

Let $G(i)=\frac{F(i)}{5}$. Then by our calculation $i(i+1)(i+2)(i+3)=G(i)-G(i-1)$.

Now consider the sum $\sum_{i=1}^n i(i+1)(i+2)(i+3)$. This is $$(G(1)-G(0))+(G(2)-G(1))+G(3)-G(2)) +\cdots+(G(n)-G(n-1)).$$ Observe the telescoping. Since $G(0)=0$, the above sum is equal to $G(n)$. Thus $$\sum_{i=1}^n i(i+1)(i+2)(i+3)=G(n)=\frac{n(n+1)(n+2)(n+3)(n+4)}{5}.$$

Exactly the same idea works in general.

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The hypothesized equality can be written as follows: for any $m$, we conjecture $$\sum^{k}_{n=1}\frac{(n+m)!}{(n-1)!}=\frac{(k+m+1)!}{(m+2)(k-1)!}$$ Dividing both sides by $(m+1)!$, we have $$\sum^{k}_{n=1}\frac{(n+m)!}{(n-1)!(m+1)!}=\frac{(k+m+1)!}{(m+2)!(k-1)!}$$ Or, in other words $$\sum^{k}_{n=1}\binom{n+m}{m+1}=\binom{k+m+1}{m+2}$$ I'm not sure how to prove this (yet), but it seems very likely that there's a neat trick for all this.

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Partial sums of sequences $a_n$ that can be expressed as polynomials in $n$ are easily found using discrete calculus.

We start with the discrete version of the Fundamental Theorem of Integral Calculus:

\begin{align*} \sum\limits_{n=1}^k a_n &= \sum\limits_{n=1}^k (\Delta b)_n \\ &= (b_2 - b_1)+(b_3 - b_2)+ \ldots + (b_k - b_{k-1}) + (b_{k+1} - b_k) \\ &= b_{k+1} - b_1 \end{align*}

where $(\Delta b)_n = b_{n+1} - b_n$ is the forward difference. Finding the partial sum has now been reduced to finding a sequence $b_n$ such that $(\Delta b)_n = a_n$.

We will find $b$, the antiderivative of $a$, using falling powers, which are defined by

$$n^{\underline{k}} = n(n-1)(n-2)\ldots (n-k+1)$$

where $k$ is an integer and, by a second definition, $n^{\underline{0}}=1$. For example

$$n^{\underline{3}} = n(n-1)(n-2).$$

We now need one more result: the discrete derivative of $n^{\underline{k}}$ is given by

\begin{align*} \Delta n^{\underline{k}} &= (n+1)^{\underline{k}} - n^{\underline{k}} \\ &= (n+1)n^{\underline{k-1}} - n^{\underline{k-1}}(n-k+1) \\ &= kn^{\underline{k-1}} \end{align*}

Let's now find the partial sum for a particular case:

\begin{align} \sum^{k}_{n=1}n(n+1)(n+2) &= \sum^{k}_{n=1} (n+2)^{\underline{3}} \\ &= \sum^{k}_{n=1} \Delta \left[\frac{1}{4} (n+2)^{\underline{4}}\right] \\ &= \frac{(k+3)(k+2)(k+1)k}{4} - \require{cancel}\cancelto{0}{\frac{(1+2)(1+1)(1-0)(1-1)}{4}} \end{align}

The general case:

\begin{align} \sum^{k}_{n=1} (n+p)^{\underline{p+1}} &= \sum^{k}_{n=1} \Delta \left[\frac{1}{p+2} (n+p)^{\underline{p+2}}\right] \\ &= \frac{(k+1+p)(k+p)\ldots [(k+1+p)-(p+2)+1)]}{p+2} \\ &= \frac{(k+1+p)(k+p)\ldots k}{p+2} \end{align}

where $p>0$ is an integer.

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[I haven't finished yet. Will be gradually improved]

If we do it this way, then probably, it will become more insightful.

$$\sum^{k}_{n=1}1=k$$

then we preserve everything from the right side exactly as it is. $$\sum^{k}_{n=1}n=\frac{k(k+1)}{1 \cdot 2}$$ [there will be a picture of the right triangle]

Now again, preserve everything from the right side: $$\sum^{k}_{n=1}\frac{n(n+1)}{1 \cdot 2}=\frac{k(k+1)(k+2)}{1 \cdot 2 \cdot 3}$$

[there will be a picture of the 6 pyramids composed into rectangular parallelepiped]

$$\sum^{k}_{n=1}\frac{k(k+1)(k+2)}{1 \cdot 2 \cdot 3}=\frac{k(k+1)(k+2)(k + 3)}{1 \cdot 2 \cdot 3 \cdot 4}$$

[There should be a description of the connection between simple combinations and combinations with repetition]

$${n \choose k} = \frac{n \cdot (n - 1) \cdot \ldots \cdot (n - k + 1)}{k \cdot (k - 1) \cdot \ldots \cdot 1} = \frac{n^{\underline{k}}}{k!}$$ $$\left(\!\middle(\genfrac{}{}{0pt}{}{n}{k}\middle)\!\right) = \frac{n \cdot (n + 1) \cdot \ldots \cdot (n + k - 1)}{k \cdot (k - 1) \cdot \ldots \cdot 1} = \frac{n^{\overline{k}}}{k!}$$

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You should use the sandbox if you are part way through an answer. –  Michael Albanese 2 days ago
@Michael: Sorry, I don't understand how to use it. Probably, I just don't have the rights to use that post or I need to do something beforehand, but I don't see [Add answer] button there and I cannot edit them neither. –  Pixar 2 days ago
Ah, you might not have enough reputation yet. Oh well, keep it in mind for future use. –  Michael Albanese 2 days ago