As I am not good at math, I would like to construct an expression having the form like:
$ a^n (\sum_{i=0}^{n-1} \lambda_i \cdot b^i ) +(\sum_{i=0}^{n-1} \lambda_i \cdot a^i)\cdot b^n + \lambda_n\cdot a^n \cdot b^n, \quad (1)$
where $\lambda_i$ is a free parameter to be determined.
My goal is to determine $\lambda_i$ such that Express (1) has a neat and simple form. $\lambda_i \ (i=1,...,n)$ can be any real positive number, but they should satsify $\lambda_i < \lambda_{i+1}$ for $i=0,...,n-1$. For example, when $n=3$,
$ 1a^3 + 6a^3b+15a^3b^2+20a^3b^3+15a^2b^3+6ab^3+1b^3.$
Currently, I choose $\lambda_{i} = {2n\choose{i}}$ which is a bionormal coefficient. But it seems that Express (1) is hard to be simplied in this case.
My question is that can anyone give me a hand to find a good coefficient $\lambda_i$ to simply Express (1) in a neat form. Thanks!