bibkey: hanna2026a395842 authors: Paul D. Hanna year: 2026 title: “OEIS A395842, diagonal coefficients of the compositional iterates of A177775” doi: null url: https://oeis.org/A395842 claim: “It appears that all terms are even after the initial term.” strata_touched:

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

OEIS A395842

Hanna’s entry dated May 20, 2026 defines a(n) as the coefficient of x^n in the n-fold compositional iterate of the generating series of A177775. That series starts x+x² and its coefficients are determined by equality of the degree-n coefficients in the n-th and (n−1)-st iterates, for n>2. The entry presents eventual evenness as a conjecture, with no proof attached. The formal theorem treats every n≥2, including a(2)=2.

Verified locator

url: https://oeis.org/A395842

The official text https://oeis.org/search?q=id:A395842&fmt=text is revision 9, dated May 23, 2026. Its DATA begins 1,2,−6,66,−1100. The entry has no DOI. The May 19, 2026 odd-coefficient comment in https://oeis.org/search?q=id:A177775&fmt=text is explicitly a conjecture and is not used as a proved premise.

Proof scope

The proof in DiagonalIterateEven is derived in this repository. Over the field with two elements, it uses the compositional inverse H of X+X². Iterate doubling and degree induction establish H’s diagonal constraints; uniqueness then identifies the reduction of the integer series with H. No proof is claimed for the separate odd-coefficient characterization of A177775, and no publication priority is claimed.