Problem 190

Maximising a weighted product

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.

加权乘积最大化

记S_m = (x₁, x₂, … , x_m)是m元正实数组，其中x₁ + x₂ + … + x_m = m，且P_m = x₁ * x₂² * … * x_m^m取得其最大值。

例如，可以验证，[P₁₀] = 4112（[ ]是截尾函数）。

对于2 ≤ m ≤ 15，求Σ[P_m]。