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Math Help - Help understanding the notation in Faulhaber's formula

  1. #1
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    Help understanding the notation in Faulhaber's formula

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

    \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 B_{1 = -\frac{1}{2}

    Unfortunately, I don't understand \left( \begin{matrix} p + 1\\ j\end{matrix}\right). Is that a vector?
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  2. #2
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    Re: Help understanding the notation in Faulhaber's formula

    Quote Originally Posted by datanewb View Post
    I was curious about the general Faulhaber's formula which states according to wiki
    \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 B_{1 = -\frac{1}{2}
    Unfortunately, I don't understand \left( \begin{matrix} p + 1\\ j\end{matrix}\right). Is that a vector?
    Usually \binom{N}{k}=\frac{N!}{k!(N-k)!}.
    Thanks from datanewb
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  3. #3
    GJA
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    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 \dbinom{3}{2}.

    The exact formula is given by

    \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!
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    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 View Post
    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 \dbinom{3}{2}.

    The exact formula is given by

    \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!
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  5. #5
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    Re: Help understanding the notation in Faulhaber's formula

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  6. #6
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    Re: Help understanding the notation in Faulhaber's formula

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