The complete A381737 Conjecture 1 enumeration

OEIS A381737, revision 28, by Mike Sheppard (2025), names the sequence:

Orders k of Hermite polynomials whose maximal coefficient in absolute value appears twice.

Its Formula field states:

Conjecture 1: a(n) = 2n(n + 2) + (n + 1)*(-1)^(n+1).

The offset is 1,1. Let $H_k$ be the physicists Hermite polynomial, with the original coefficients represented by HermiteSignedMode.physicistsHermite over the rationals. For natural $k$ and $j$, write

$$ M(k,j)\iff j\le k\ \land\ \forall i\le k,\quad |[x^i]H_k|\le |[x^j]H_k|. $$

Exactly two maxima means that there exist $u<v$ such that, for every natural $j$, $M(k,j)$ holds if and only if $j=u$ or $j=v$. The entire row from zero through $k$ is compared, including zero coefficients of the other parity. For every positive natural index, define the integer value

$$ a(n)=2n(n+2)+(n+1)(-1)^{n+1}. $$

The complete assertion is that $a(n)>0$ for every $n\ge1$, that $a(n)<a(m)$ whenever $1\le n<m$, and that for every natural order $k$, including zero, exactly two maxima occur if and only if there is exactly one $n\ge1$ with $(k:\mathbb Z)=a(n)$. The additions and products in the formula are integer operations; $n+1$ in the exponent is natural. There is no finite cutoff or asymptotic qualification.

This dossier concerns the whole external Conjecture 1, preregistered at https://github.com/the-omega-institute/trureturing/issues/25855#issuecomment-6590328488. The full-row meaning follows the source name and its physicists $H_8$ example. The neighboring absolute-mode result supplies coefficient maximality, while the tie classification and enumeration identify every qualifying order. Conjectures 2 and 3 are outside this resolution.

The Library note identifies revision 28, the Name and Formula fields and the one-based offset. Programs and finite examples do not supply the universal enumeration. The needed relation is an exact tie criterion for the original coefficient row, followed by an exhaustive solution of that criterion and uniqueness of the positive index. The source locator does not certify worldwide current openness or priority, and no such claim is made here.

For $2r\le k$, let $w(k,r)$ be the absolute value of the original coefficient at exponent $k-2r$. The existing absolute-mode results give positivity, the adjacent coefficient identity, maximality at $\operatorname{mode}(k)$ and strict decrease beyond that index. The strict threshold before the last adjacent pair restricts a maximizing index to $\operatorname{mode}(k)$ or $\operatorname{mode}(k)-1$. A tie occurs exactly when, for some $r\ge1$ with $2r\le k$,

$$ (k-2r+1)(k-2r+2)=4r. $$

Conversely this equality forces $r=\operatorname{mode}(k)$ and precisely two maxima. The coefficient extraction theorem transfers this statement to every exponent in the full polynomial row; parity zeros cannot maximize a positive mode weight.

Put $j=k-2r$. The equality forces $j=4t+2$ or $j=4t+3$. These give, respectively, $k=8t^2+18t+8=a(2t+1)$ and $k=8t^2+22t+13=a(2t+2)$, and both families supply a valid $r$. Every positive index is of exactly one of these forms. The adjacent integer difference $a(n+1)-a(n)$ is $2n+3$ or $6n+9$, hence positive; $a(1)=8$. Strict increase supplies uniqueness as well as positivity and exhaustion.

A positive index with a nonpositive value, a pair $1\le n<m$ with $a(n)\ge a(m)$, or a natural order at which exactly two full-row absolute maxima disagree with the unique-positive-index condition would falsify the complete event. A finite list, a different Hermite normalization or a restriction to nonzero coefficients cannot replace this criterion.

D5/S3/Analytic/Polynomial/HermiteAbsoluteTieOrders.result proves the complete claim, whose definitions retain the original row, exactly two distinct maximizers, every positive index and every natural order. Its owning Scribe binds one typed OpenProblemResolutionClaim of kind Proved to this slug. The mathematical sources and their Reg evidence are reused unchanged.

Every child theorem is conservatively classified proof_shape: bind-only: max_iff_weight, strict_before, equal_mode_indices, adjacent_equal, tie_is_mode, tie_characterization, a_odd, a_even, a_strictMono, a_positive, tie_orders and result. The admission basis is open-problem-resolution; escape_witness: none and utility: none. This classification grants no new abstract mathematical-content credit. The helper proofs support the complete external resolution.

The source classification is theorem. The resolution covers the complete Conjecture 1 enumeration, including positivity, strict increase, exhaustive classification and unique positive indexing. Conjectures 2 and 3, other normalizations and new utility claims remain outside its scope.

The interpretation of the source's name, offset and formula as the explicit quantified statement above is a source-to-formal correspondence; the Lean kernel checks the formal statement and does not interpret OEIS prose. The Library locator is citation-only. No exhaustive literature search, worldwide current-openness, originality or priority certification follows from this dossier. Finite examples and source programs are not universal proof evidence.