bibkey: hanna2024a376527 authors: Paul D. Hanna year: 2024 title: “OEIS A376527, square sums of Carlitz-Riordan q-Catalan rows and parity conjecture” doi: null url: https://oeis.org/A376527 claim: “Conjecture: a(n) is odd iff n = 2^k - 1 for some k >= 0.” strata_touched:

• D5/S1/Recurrence/Parity/QCatalanSquareSumParity license: citation-only triage: anchor

OEIS A376527

Paul D. Hanna’s comment of October 11, 2024 records the parity conjecture quoted above. The entry defines

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

where A227543 is the Carlitz-Riordan q-Catalan triangle. Its row polynomials are determined by the generating-function equation

A(x,q) = 1 + x*A(q*x,q)*A(x,q).

Writing A(x,q) = Sum_n R_n(q)*x^n and comparing the coefficient of x^(n+1) gives

R_0(q) = 1 and R_(n+1)(q) = Sum_{i+j=n} q^i*R_i(q)*R_j(q).

At q=1, the recurrence becomes the Catalan convolution. Consequently the sum of a row is the nth Catalan number. In characteristic two each coefficient satisfies u^2=u, so the row’s square sum has the same parity as its row sum. The shifted Catalan series therefore places the odd indices exactly one below a power of two.

The directed formal dependency is D5/S1/Recurrence/Parity/QCatalanSquareSumParity to D5/S1/Recurrence/Invariants/CatalanCompositionSquareParity, through catalan_unique for the row-sum identification and binary_catalan for the binary support.

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• URL: https://oeis.org/A376527