# How to calculate sum of sum of subsets using polynomial multiplication?

Recently I was reading about polynomial multiplication, and came across solution of one interesting problem.

Which is finding sum of product of all $k-subsets$.

which is as following:

if we have set $[2,3,4]$. we can construct polynomial $(x-2)(x-3)(x-4)$. and for answer for $k-subset$ is absolute value of coefficient of $x^{n-k}$, where $n$ is size of subset.

Now I have few more question and I want to know how can i find the solution of them.

1> Find $\sum\limits_{k-size-subsets}Fibonacci(\prod\limits_{i \in subset} i)$. here if we want to manipulate the product by applying some function before adding.

2> Find$\sum\limits_{k-size-subsets} Fibonacci(\sum\limits_{i \in subset} i)$. Same way If i want to find sum of $k-subset$ elements, apply some function on it and sum of all that $k-subsets$.

• Your setup is fairly clear, but your actual questions are not. Please clarify. – Rory Daulton Jun 10 '15 at 11:32
• If you mean : $$u_n=\sum_{A\in [1,n]}\prod_{i\in A} i$$ then you just need to know that $u_{n+1}=(n+1)u_n+u_n$ and from here you can get the answer – Elaqqad Jun 10 '15 at 11:36
• Just wait I will update that text with mathematical expression. – CodeLover Jun 10 '15 at 11:38
• I hope it's clear now. – CodeLover Jun 10 '15 at 11:44