slug: oeis-a398690-palindromic bibkey: chapoton2026a398690 doi: null url: https://oeis.org/A398690 triage: theorem motivation_gids:
- D5/S3/Analytic/OeisA398690Palindromic
Palindromic numerators of OEIS A398690
Problem
OEIS A398690 defines A(r,q) by its signed finite sine sum. For every
n >= 0, let F_n(z) = sum_{q>=0} A(n+3,q) z^q. Its comment says that
F_n has denominator (1-z)^(n+1) and conjectures that its numerator
is palindromic. The formal result proves that (1-z)^(n+1) F_n(z) is a
polynomial of degree exactly n and that its coefficients at j and
n-j agree for every 0 <= j <= n.
Motivation
Chapoton’s 2026 entry explicitly labels the all-row palindromicity assertion a conjecture. Preregistration issue #10928 fixes the source formula, its row shift, and the universal target before this proof.
Gap
The OEIS comment provides the rational-row claim but no proof of coefficient reversal. The prior project triage records only finite rows. The A107735 cross-reference uses a different even-row normalization.
Route
Chebyshev factorization gives a product over the sine nodes. Formal
logarithmic differentiation relates its inverse power sums to the
coefficients of an odd Chebyshev polynomial. The derivative recurrence
builds a degree-bounded polynomial model in 2q+1, with parity matching
the row. Composing with 2X+1 turns parity into reflection about -1/2.
A binomial-basis calculation then turns this reflection into coefficient
reversal of the generating-series numerator. The source sine sum is
identified with the model for every q.
Falsifier
A row and coefficient index violating reversal for the stated sine sum would refute the conjecture. A finite list of matching rows cannot establish the universal claim.
Evidence
The source definition and the quantified result are in
D5/S3/Analytic/OeisA398690Palindromic.lean. The OEIS revision and
normalization are recorded in Library/Analytic/chapoton2026a398690.md.
Preregistration: issue #10928. The fresh Lean report records the source
owner’s Reg/D5/S3/Analytic/OeisA398690Palindromic.lean registration as
declared_validated with evidence reference
31acfe53a65b588e9ffa2a3532b41a112d1e28b7ed36528f574b19c3fa702d49.
Its escape residual is open; this does not assert inexhaustibility.
The source result and registration use only propext, Classical.choice,
and Quot.sound.
Triage
theorem. The result proves the all-row source conjecture.
ASSUMED-UNVERIFIED
The source search was bounded. Mukai (2003), p. 483 was not inspected; priority over a possible general reciprocity result there is unverified. The claim concerns the simplified A398690 array, not the even-row normalization of A107735.