Solving a system of equations in $\mathbb{N}^{n+1}$

Question

I post here because I'm struggling doing something. Actually, I wanted to determine, for $n \in \mathbb{N}_{\geq 1}$, the pair of $(n+1)$-tuples $(a_0, \dots, a_n)$, $(b_0, \dots, b_n) \in \mathbb{N}^{n+1}$ such that :

$$ a_0 + \dots + a_n = b_0 + \dots + b_n$$

and

$$ a_1 + 2a_2 + 3a_3 + ... +na_n = b_1 + 2b_2 + 3b_3 + ... +nb_n $$

Is there an easy way to do it ? (I've really struggle and didn't succeed to prove what I wanted to because of this, so I'm getting a little bit frustrated)

Thank you !

Answer 1

There are lots of them. You have two equations in $2n+2$ unknowns. Pick all the variables except $a_0,b_0,a_1, b_1$ however you like. Add up all the terms of the second equation except $a_1,b_1$. Whichever side is larger, make its last variable whatever you like and compute the last one on the other side. Now do the same for the first equation. This allows you to pick $2n$ of the variables however you like.