I came up with this formula for compound summation but don't know how to represent it correctly. Is there a standard way to represent an arbitrary level of compound summation where left sums depend on the sum directly to their right which depends on the sum to their right etc...?(adsbygoogle = window.adsbygoogle || []).push({});

level: 1

[tex]

\sum_{j=0}^m a + cj = \frac{(m + 1)(2a + cm)}{2}

[/tex]

level: 2

[tex]

\sum_{j=0}^m \frac{(j + 1)(2a + cj)}{2} = \frac{(m + 1)(m + 2)(3a + cm)}{6}\\

[/tex]

level: 3

[tex]

\sum_{j=0}^m \frac{(j + 1)(j + 2)(3a + cj)}{6} = \frac{(m + 1)(m + 2)(m + 3)(4a + cm)}{24}\\

[/tex]

level: 4

[tex]

\sum_{j=0}^m \frac{(j + 1)(j + 2)(j + 3)(4a + cj)}{24} = \frac{(m + 1)(m + 2)(m + 3)(m + 4)(5a + cm)}{120}\\

[/tex]

level: n

[tex]

= \frac{(m + n)!(a(n + 1) + cm)}{m!(n + 1)!}

[/tex]

For level "n" would something like this be correct?

[tex]

\sum_{j=0}^m \sum_{j=0}^m ... \sum_{j=0}^m a + cj = \frac{(m + n)!(a(n + 1) + cm)}{m!(n + 1)!} = a(n + 1) + cm \prod_{j=1}^n \frac{m + j}{j + 1}

[/tex]

Or is there a way to explicitly specify how many levels (n) of compound summation are desired?

Thanks

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# Help with writing compound sum

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