Maximising a weighted product (noch nicht übersetzt)

Let S_m = (x₁, x₂, ... , x_m) be the m-tuple of positive real numbers with x₁ + x₂ + ... + x_m = m for which P_m = x₁ * x₂² * ... * x_m^m is maximised.

For example, it can be verified that [P₁₀] = 4112 ([ ] is the integer part function).

Find Σ[P_m] for 2 ≤ m ≤ 15.