# Sum of the products of combinations from a set

I'm struggling to work out how to write what it is that I need to write in neat mathematical notation.

Given a set of numbers $$A = \{a_1,a_2,\dots,a_n\}$$, I need write an expression

$$f(a_h) = a_h\left(1+ \sum_{i=h+1}^n \left(a_i+a_i\sum_{j=i+1}^n \left( a_j + a_j \sum_{k=j+1}^n\dots\right)\right)\right),$$

so that I am summing everything from $$a_1a_2$$ through to $$\prod_{i\neq2}^na_i$$. Obviously, I can write it as this summation in a summation in a summation but I don't think it is particularly clear or neat as an expression. So, my question is whether there is a clearer and tidier way of writing this function or if I need to keep the expression as is and just live with it.

For clarity, because I'm not convinced that I am clear, if I had the set $$A=\{a,b,c,d\}$$, $$f(a)=a+ab+ac+ad+abc+abd+acd+abcd$$ $$f(b)=b+bc+bd+bcd$$

• If you look at the example, you can write $f(a) = a(1 + f(b) + f(c))$. I think you can show that $f(a_i) = a_i(1 + \sum_{j=i+1}^{n-1} f(a_j))$ Apr 29, 2021 at 14:53

If you look at the example, you can write $$f(a) = a(1 + f(b) + f(c))$$.
You can show that $$f(a_i) = a_i(1 + \sum_{j=i+1}^{n-1} f(a_j))$$ by induction. Consider $$a_k,...,a_n$$ holds the formula, and show for $$a_{k-1}$$.
Apparently you want the sum of products over all subsets of $$A$$ that contain $$a_h$$ as their first element. That is $$a_h(1+a_{h+1})(1+a_{h+2})\ldots(1+a_n)$$