Pro Landscape 19 Full Crack ((NEW))
Pro Landscape 19 Full Crack ((NEW))
Pro Landscape 19 Full Crack
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Seeking an identity given the order of summation
As I understand it,
$$
\sum_{n=1}^{k}\sum_{m=1}^{n}f(n,m) = \sum_{n=1}^{k}\sum_{m=1}^{n}f(n,m)+\sum_{n=1}^{k}\sum_{m=1}^{k}f(n,m)+\ldots +\sum_{n=1}^{k}\sum_{m=1}^{k}f(n,m)
$$
I was wondering, is there a closed form for this summation without simply having to write it out?
A:
Let $S_1, S_2, \ldots S_k$ be a permutation of the numbers $\{1, 2, \ldots, k\}$. We can write $f(n,m) = g(n-m)$ where $g$ is a function that is periodic in $n$. Then
$$
\sum_{n=1}^{k}\sum_{m=1}^{n}f(n,m) = \sum_{n=1}^{k}\sum_{m=1}^{n}g(n-m) = \sum_{i=0}^{k-1}\sum_{n=1}^{k}g(n-i) = \sum_{i=0}^{k-1}\sum_{n=i}^{k-i}g(n) = \sum_{i=0}^{k-1}\sum_{n=0}^{k-i}g(n) = \sum_{i=0}^{k-
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