lean_workbook_plus_29911

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Proved status carries no information about that problem. Do not import this node or use it as a dependency.

This suggests recursively computing the $S_k$ . Hence $S_1(n) = (n+3)\binom{2n+1}{n} - S_0(n+1) = \frac{n(n+1)}{n+2} \binom{2n+1}{n},$ and \n$S_2(n) = (n^2-2n-5)\binom{2n+1}{n} - S_0(n+1) - 2S_1(n+1) \ = \frac{n(n+1)^3}{(n+2)(n+3)}\binom{2n+1}{n}.$

Why this node was retired

The posted statement is

theorem lean_workbook_plus_29911  (n : ℕ)
  (S : ℕ → ℕ)
  (h₀ : S 0 = 0)
  (h₁ : ∀ n, S (n + 1) = (n + 3) * (2 * n + 1).choose n - S n)
  (h₂ : ∀ n, S (n + 2) = (n^2 - 2 * n - 5) * (2 * n + 1).choose n - S n - 2 * S (n + 1)) :
  S 2 = (n * (n + 1)^3 / ((n + 2) * (n + 3))) * (2 * n + 1).choose n   :=  by sorry

The fragment omits the original sum definition, and its S₂ coefficient may itself need checking before a complete corrected formula is asserted.

A proof of a malformed proposition can be a correct proof of that proposition, so this is not a judgment on the accepted submission — but the Proved status must not be read as settling the problem shown above.

Proposed corrected statement

Restore the two-index quantities S(k,n), their original sum definitions and recurrences, using integer or rational arithmetic for alternating terms; do not replace them by incompatible recurrences for one sequence S.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.