bibkey: hanna2026a396795 authors: Paul D. Hanna year: 2026 title: “OEIS A396795, g.f. satisfying A^2(x) A^3(x) = x^2 + 25 x^3” doi: null url: https://oeis.org/A396795 claim: “G.f. A(x) satisfies A^2(x) * A^3(x) = x^2 + 25*x^3, where A^n(x) is the n-th iteration of A(x). Conjecture: a(n) == 0 (mod 5) for n > 1.” strata_touched:

• D5/S1/Recurrence/Residue/IterateProductTwentyFiveModFive license: citation-only triage: anchor

OEIS A396795

Paul D. Hanna’s entry, dated June 6, 2026, specifies the generating equation and coefficient conjecture quoted above. The superscripts on A denote compositional iteration, not ordinary powers. The normalized formal series has integer coefficients, constant coefficient zero, and coefficient one at degree one; a(n) is its degree-n coefficient. The OEIS offset is 1,2.

The module proves square-zero lifting for a variable natural modulus m: if f is congruent coefficientwise to X modulo m and m divides i+j, the product of its i-th and j-th compositional iterates is congruent to X^2 modulo m^2. It applies this theorem with (m,i,j)=(5,2,3). Successive exact coefficient corrections construct the normalized integer solution of the displayed equation. A coefficient perturbation argument proves uniqueness and divisibility of every coefficient above degree one by five.

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• URL: https://oeis.org/A396795