Prove Pascal's Triangle-type Function - Discrete Mathematics

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Homework Statement

For all n ∈ Z⁺, the function P_n of i variables is defined recursively as follows:
P_n(x₁,...,x_n) = P_n-1(x₁ + x₂, x₂ + x₃,...,x_n-1 + x_n) and P1(x₁) = x₁.
Find a closed formula for P_n.

Homework Equations

P_n(x₁,...,x_n) = P_n-1(x₁ + x₂, x₂ + x₃,...,x_n-1 + x_n) and P1(x₁) = x₁.

The Attempt at a Solution

So far, I've found it definitely follows a Pascal's Triangle-esque pattern:

P₁(x₁) = x₁
P₂(x₁,x₂) = P₁(x₁ + x₂) = x₁ + x₂
P₃(x₁,x₂,x₃) = P₂(x₁ + x₂, x₂ + x₃) = P₁(x₁ + 2x₂ + x₃) = x₁ + 2x₂ + x₃
P₄(x₁,x₂,x₃,x₄) = P₃(x₁ + x₂, x₂ + x₃, x₃ + x₄) = P₂(x₁ + 2x₂ + x₃, x₂ + 2x₃ + x₄) = P₁(x₁ + 3x₂ + 3x₃ + x₄) = x₁ + 3x₂ + 3x₃ + x₄

I know it's very similar to Pascal's triangle, but I'm just not sure how to find a closed form for it.

I'd appreciate any help; thanks!

 

[Char. Limit]

That's pascal's triangle, but that also has a lot to do with the binomial theorem. Try using that.