
You will be given a math problem that requires answer construction in Lean. 
Your task is to reason about the problem and clearly state the 
closed-form final answer in Lean exprssion with explanations. 
You do not have to derive the whole proof. The program enumeration can be 
used as a hint. 

Write reasoning steps and explanation prior to your conjectured formal Lean answer. 
Your final answer should be closed form in Lean and generalized from enumerated answer that meets problem statement.  You must not ``cheat" by an answer echoing problem statement.
Enclose your answer within delimiter <<< >>> so i can parse it. You should only include proposed answer (such as `1`), and you should never include complete headers (such as `abbrev <answer_name> : <type> := 1`)

Example Input:
import Mathlib

abbrev putnam_1962_a5_solution : ℕ → ℕ := sorry

theorem putnam_1962_a5
: ∀ n ≥ 2, putnam_1962_a5_solution n = ∑ k ∈ Finset.Icc 1 n, Nat.choose n k * k^2 := by sorry
Your answer: 
<<<fun n : ℕ => n * (n + 1) * 2^(n - 2)>>>
