Lets say we have a set {1, 2, 3, 4, 5}, There are 32 subsets. Now if we take these subsets and arrange them according to the sum of their members; For eg: The subset {2, 3} would be in the same group as the subset {1, 4} as the sum of their members is the same. If we use the "Generating function" as used in this video: https://www.youtube.com/watch?v=bOXCLR3Wric, We get the number of such arrangements as the coefficients of a long polynomial.
Generating function: f(x) = $(1+x^1)(1+x^2) .... (1+x^n)$, where n is the number of elements in the set. while expanded it forms: $1+x^1+x^2+2x^3+2x^4+3x^5+4x^6+5x^7 + ....$
Now is it possible to represent this function neatly with a single equation using Binomial theorem (Binomial coefficients?)
Given $[n]=\{1,2,\ldots,n\}$ the wanted generating function is \begin{align*} f(x)=\prod_{j=1}^n\left(1+x^j\right)\tag{1} \end{align*} since each of the elements in $[n]$ is either an element of the subset under consideration or not an element. The coefficients of $x^j$ in $f$ are for increasing $n$: \begin{align*} \begin{array}{lll} n=0:&\{\emptyset\}&\color{blue}{1}\\ n=1:&\{\emptyset,\{1\}\}&\color{blue}{1,1}\\ n=2:&\{\emptyset,\{1\},\{2\},\{1,2\}\}&\color{blue}{1,1,1,1}\\ n=3:&\{\emptyset,\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\}\}&\color{blue}{1,1,1,2,1,1,1}\\ \dots&\ldots&\ldots \end{array} \end{align*} The blue marked sequence of coefficients of $x^n$ of increasing $n$ is stored in OEIS as A053632. Note, there is no other simple representation given.