I am looking for information about sums of the form:
$$\sum_{i=1}^n \sum_{j=1}^i f(j)$$
But not just that form, but arbitrarily many stacked sums. Even just a name would help. To be specific about what I mean, I have been using the following definition:
$$(\sum_{i=a}^b)^1f(i) := \sum_{i=a}^bf(i)$$ $$(\sum_{i=a}^b)^xf(i) := \sum_{i=a}^b (\sum_{j=a}^i)^{x-1} f(j)$$
I am particularly interested in:
$$f_x(n) = (\sum_{i=1}^n)^x1$$
Any pointers to information on any of these ideas would be greatly appreciated. Including a name for this, better notation, any interesting solutions involving these sums etc.
I don't know a name for those sums, but I surely know you can calculate them easily:
$$ (\sum_{i=a}^b)^xf(i) = \sum_{i=a}^b\binom{b-i+x-1}{x-1}f(i) $$
It's a double counting obtained from the question "how many time I add $f(i)$?".
In fact, if you have x variables $i=i_1\le i_2\le i_3\le \dots \le i_x$, you have an addend $f(i)$ for all choices of the values of the variables distinct from $i$.
So you have to count the number of ordinated $(x-1)$-uples of integers between $i$ and $b$.
It's a well-known formula (there's in every combinatoric book you may find), that the number of those is
$$ \binom{b-i+x-1}{x-1} $$ So you have the desired coefficients.
In particular,
$$ (\sum_{i=a}^b)^x1 = \sum_{i=a}^b\binom{b-i+x-1}{x-1}=\binom{b-a+x}{x} $$