slug: oeis-a376527-q-catalan-square-sum-parity bibkey: hanna2024a376527 doi: null url: https://oeis.org/A376527 triage: theorem motivation_gids:

• D5/S1/Recurrence/Invariants/CatalanCompositionSquareParity.catalan_unique
• D5/S1/Recurrence/Invariants/CatalanCompositionSquareParity.binary_catalan

Parity of q-Catalan row square sums

Problem

OEIS A376527 defines

a(n) = Sum_{k=0..n*(n-1)/2} A227543(n,k)^2.

Paul D. Hanna’s comment of October 11, 2024 states, verbatim:

Conjecture: a(n) is odd iff n = 2^k - 1 for some k >= 0.

Here A227543 is the Carlitz-Riordan q-Catalan triangle defined by A(x,q) = 1 + x*A(q*x,q)*A(x,q).

Motivation

The assertion covers every natural row index, including zero. For the first fifteen listed values, the odd indices are 0, 1, 3, 7; this is supporting evidence rather than a proof of the universal statement.

Gap

Repository searches found no A376527 declaration. Pinned Mathlib contains no Carlitz-Riordan q-Catalan development. The repository does contain the shifted Catalan equation, its uniqueness theorem, and its binary support theorem, but no typed bridge from q-Catalan rows to that series.

Route

Let R_n(q) be the nth row polynomial. Coefficient extraction from the defining bivariate equation gives R_0=1 and R_(n+1)=Sum_{i+j=n} q^i R_i R_j. This recurrence is used directly as the row-polynomial definition.

Evaluation at q=1 gives the Catalan convolution for the row sums. After a one-place shift, their generating series has zero constant coefficient and satisfies F=X+F^2. The established uniqueness theorem identifies it with the shifted Catalan series.

The degree of R_n is at most n*(n-1)/2, so evaluation at one is exactly the finite sum across the A227543 row. Modulo two, u^2=u term by term. Thus the square sum reduces to the row sum, and the established binary Catalan theorem places coefficient one precisely at indices 2^k. Accounting for the series shift gives n=2^k-1.

Falsifier

A natural row index whose square sum has oddness different from membership in {2^k-1 : k >= 0} would contradict the theorem.

Evidence

• Lean module: D5/S1/Recurrence/Parity/QCatalanSquareSumParity.lean.
• Resolution anchor: hanna_conjecture.
• Row recurrence: qCatalanRow_succ.
• Catalan bridge: rowSum_eq_catalan.
• Parity companion: squareSum_mod_two_eq_catalan.
• Directed dependencies: rowSum_eq_catalan to catalan_unique, and hanna_conjecture to binary_catalan through the parity companion.

The exact final statement is:

theorem hanna_conjecture (n : ℕ) :
    Odd (squareSum n) ↔ ∃ k : ℕ, n = 2 ^ k - 1

Triage

theorem. The formal proof closes the universal parity assertion recorded by OEIS.

ASSUMED-UNVERIFIED

The source wording, attribution, date, and numerical check were supplied by the implementation brief. Source-to-Lean identification and publication priority are not kernel-checked facts.