I look for formula that is expand the n-th product $\prod_{j=1}^{n}(a_{j}+b_{j})$ to sum. I know this sum includes $2^{n}$ terms, but I can't write this expand as a simple formula. I searched in the "Handbook of Mathematics, I.N. Bronshtein" but I didn't find the desired formula.
Let me suggest the following: $\sum_{\{(x_1,\ldots,x_n) : x_i \in \{a_i,b_i\}, i = 1, \ldots,n\}} x_1\cdot x_2 \cdot \ldots \cdot x_n$.
Alternatively, you could also first define a set $S := \{a_1,b_1\} \times \{a_2,b_2\} \times \ldots \times \{a_n,b_n\}$ and write $\prod_{j=1}^{n}(a_{j}+b_{j}) = \sum_{(x_1,\ldots,x_n) \in S} x_1\cdot x_2 \cdot \ldots \cdot x_n$.
Write the first several products, each fully expanded, e.g.:
P(1) = a(1) + b(1)
P(2) = a(1)*a(2) + a(1)*b(2) + b(1)*a(2) + b(1)*b(2)
etc. and notice the general pattern: Each term of the sum consists of a permutation of n occurrences of a and b, with multiplication and the arguments (1), (2), etc. inserted in that order in each permutation. Your desired sum is then the sum of all such permutations. E.g. for n=3:
a(1)*a(2)*a(3)
a(1)*a(2)*b(3)
a(1)*b(2)*a(3)
a(1)*b(2)*b(3)
b(1)*a(2)*a(3)
b(1)*a(2)*b(3)
b(1)*b(2)*a(3)
b(1)*b(2)*b(3)
P(3) is the sum of the above terms. Etc. for n=4, …
Other than a summation of a summation of a summation … etc., I know of no common form for expressing such a sum, but one could invent one. It might or might not be worthwhile to do so, depending upon what you want to do with an expression in summation form. The usual product notation will be at least as useful, for most, if not all, purposes.
It is difficult to find the general formula for n but I recommend you mathematical software Mathematica or simplified engine which you can find here:
http://www.wolframalpha.com/
Type for example the following command
Expand[Product[(a[j] + b[j]), {j, 1, 4}]] and press Enter.
You can change 4 for required number and find other solution
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