Help understanding the notation in Faulhaber's formula

I was curious about the general Faulhaber's formula which states according to wiki

$\displaystyle \sum_{k=1}^{n}k^p = \frac{1}{p+1}\sum_{j=0}^{p}(-1)^j \left( \begin{matrix} p + 1\\ j\end{matrix}\right)B_{j}n^{p+1-j},$ where $\displaystyle B_{1 = -\frac{1}{2}$

Unfortunately, I don't understand $\displaystyle \left( \begin{matrix} p + 1\\ j\end{matrix}\right)$. Is that a vector?

Re: Help understanding the notation in Faulhaber's formula

Quote:

Originally Posted by

**datanewb** I was curious about the general Faulhaber's formula which states according to

wiki
$\displaystyle \sum_{k=1}^{n}k^p = \frac{1}{p+1}\sum_{j=0}^{p}(-1)^j \left( \begin{matrix} p + 1\\ j\end{matrix}\right)B_{j}n^{p+1-j},$ where $\displaystyle B_{1 = -\frac{1}{2}$

Unfortunately, I don't understand $\displaystyle \left( \begin{matrix} p + 1\\ j\end{matrix}\right)$. Is that a vector?

Usually $\displaystyle \binom{N}{k}=\frac{N!}{k!(N-k)!}$.

Re: Help understanding the notation in Faulhaber's formula

Hi, datanewb.

The term you're asking about is read 'p+1 choose j.' If you've ever learned about permutations and combinations before, it's a combination - specifically how many ways can you pick j things from a pile containing p+1 things if order does NOT matter.

For example, consider the collection {a,b,c}. If we wanted to know how many different ways we could pick 2 things from this set it would be $\displaystyle \dbinom{3}{2}$.

The exact formula is given by

$\displaystyle \dbinom{n}{k}=\frac{n!}{k!(n-k)!}.$

If you want to know where the formula comes from let me know. Does this answer your question? Let me know.

Good luck!

Re: Help understanding the notation in Faulhaber's formula

Thank you both: Plato for you quick response and GJA for adding those excellent details. Knowing how to pronounce n choose k was very helpful, as well as it's application. I am curious where it comes from. You've already been extremely helpful! I will read up on the binomial coefficient, but if you have a good explanation, that wouldn't take too long to write, I'd appreciate hearing that as well, @GJA.

Thank you again.

Quote:

Originally Posted by

**GJA** Hi, datanewb.

The term you're asking about is read 'p+1 choose j.' If you've ever learned about permutations and combinations before, it's a combination - specifically how many ways can you pick j things from a pile containing p+1 things if order does NOT matter.

For example, consider the collection {a,b,c}. If we wanted to know how many different ways we could pick 2 things from this set it would be $\displaystyle \dbinom{3}{2}$.

The exact formula is given by

$\displaystyle \dbinom{n}{k}=\frac{n!}{k!(n-k)!}.$

If you want to know where the formula comes from let me know. Does this answer your question? Let me know.

Good luck!

Re: Help understanding the notation in Faulhaber's formula

Re: Help understanding the notation in Faulhaber's formula

Thanks, yes, that's the page I am reading. More than enough information there. :)