Metadata
bibkey: hanna2026a397594
authors: Paul D. Hanna
year: 2026
title: "OEIS A397594: quadrupled linear rows and dyadic parity"
doi: null
url: https://oeis.org/A397594
claim: "Conjecture: for n > 3, a(n) is odd iff n is of the form 2^k-1 or 2^k+1 (k > 1)."
strata_touched:
- D5/S1/Recurrence/LinearRows/QuadrupledLinearExponentDyadicSupport
license: citation-only
triage: anchor
Verified locator
Source: https://oeis.org/A397594
DOI: null. The source is OEIS A397594, revision 15; it has no arXiv identifier or pagination.
Source statement
The NAME field reads:
G.f. A(x) such that the sum of the first n coefficients in A(x/n)^n equals 4^(n-1) for n >= 1.
The FORMULA field reads:
G.f. A(x) = Sum_{n>=0} a(n)x^n has the following properties. (1) 4^(n-1) = Sum_{k=0..n-1} [x^k] A(x/n)^n for n >= 1. (2) (4n)^(n-1) = [x^(n-1)] A(x)^n / (1 - n*x) for n >= 1.
The COMMENTS field reads:
Conjecture: for n > 3, a(n) is odd iff n is of the form 2^k-1 or 2^k+1 (k > 1).
The offset is zero and a(0)=1. The formal series has integer coefficients;
its rational rescaling is PowerSeries.rescale ((m : ℚ)⁻¹) (A.map (Int.castRingHom ℚ)). The first m coefficients are those at
indices 0 through m−1. Powers are ordinary formal-series multiplication.
The quantified claim includes existence of an integer source series and
the parity equivalence for every integer series satisfying (1).
Scope
Denominator clearing connects (1) and (2). Integer triangular normalization gives forcing 4^n(n+1)^(n−1) in positive degree n. Coefficientwise iteration constructs an integer source, and reduction modulo two identifies its coefficients with the homogeneous dyadic-support solution. The statements for the related reversion series B, arbitrary even parameters, full triangle integrality, positivity and asymptotics are outside this result.
The finite arithmetic check is in the-omega-institute/trureturing-experiments,
commit b2b8f61f913caa4dbe8eace3d5ab8dae15cd5225, at
docs/reports/quadrupled-row-dyadic-parity/check.py, SHA-256
665de37ae789c10df0fd0615fd2967a95ca76e3da0997563f2252547c03a51ba.
Running python3 docs/reports/quadrupled-row-dyadic-parity/check.py exits 0
with ALL_CHECKS_OK: source coefficients agree through degree 24, formula
(1) holds through n=39, formula (2) holds through n=100, and the odd indices
from 4 through 100 are 5, 7, 9, 15, 17, 31, 33, 63 and 65. This finite check
is separate from the all-index theorem.
The literature screen is the bounded screen recorded in issue #22490. Its completeness is ASSUMED-UNVERIFIED: OpenAlex and Semantic Scholar were rate-limited, and the broad Crossref screen and two classical background works do not establish absence of an earlier settlement.