bibkey: hanna2026a397592
authors: Paul D. Hanna
year: 2026
title: "OEIS A397592: doubled linear rows and dyadic parity"
doi: null
url: https://oeis.org/A397592
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/DoubledLinearExponentDyadicSupport
license: citation-only
triage: anchor

OEIS A397592

The NAME reads:

G.f. A(x) such that the sum of the first n coefficients in A(x/n)^n equals 2^(n-1) for n >= 1.

The FORMULA specifies A(x)=Sum_{n>=0} a(n)*x^n and both equations:

(1) 2^(n-1) = Sum_{k=0..n-1} [x^k] A(x/n)^n for n >= 1.

(2) (2n)^(n-1) = [x^(n-1)] A(x)^n / (1 - nx) for n >= 1.

The original offset is 0 and a(0)=1. For an integer formal series A, A_Q means A.map (Int.castRingHom Rat), the coefficientwise map to Q[[X]]. The literal source sums coeff j((rescale(1/m,A_Q))^m) over j=0,…,m-1 for every natural m>0 and sets this rational sum equal to 2^(m-1). Powers are ordinary series multiplication. The row formula is source-attested; using it in a proof requires its equivalence to this rescaling equation.

The sole conjecture is the quoted parity claim. Its author field is Paul D. Hanna, Jul 10 2026. The entry’s b-file link covers n=0,…,400; the asymptotic formula is not a parity proof.

A397590 states a(n)=Sum_{k=0..n} 2^k T(n,k) for this y=2 series, and A(X,0)=1-F(X), where F is A397591’s series. F has zero constant term, offset 1, and the different equation [X^(m-1)](1-F)^m/(1-mX)=0 for m>1. The existing A397591 endpoint is not an identification of the A397592 source. The source relations alone do not prove integer coefficients for the entire parameter triangle.

Verified locator

url: https://oeis.org/A397592

doi: null

Revision: %I A397592 #18 Jul 11 2026 03:34:44. The complete mirror entry is at https://raw.githubusercontent.com/oeis/oeisdata/ba8580c292f2c51dd0dc3ae2f3ba052c7f17c7d3/seq/A397/A397592.seq (3,990 bytes; SHA-256 ec2009bc22f7209c0403adc0c2909bffbc24016ef40c9eb8b0c131585c8e51d8). The live OEIS text snapshot recorded in #8006 returned HTTP200 and contains that full entry (4,200 bytes; SHA-256 fef496e6b4d2dbe11bbfcaf709ed788776b6c51502c991f28528cdaf09388c86).

Proof scope

The mathematical statement couples unique existence over Z[[X]] for the literal source with the parity iff for every source solution and every n>3, using a natural k>1 and both dyadic alternatives. OEIS attests the definition and conjecture, not a proof. The corresponding problem is oeis-a397592-dyadic-parity. Its ASSUMED-UNVERIFIED section specifies the bounded literature search: exact index zeros do not establish global absence; incomplete broad queries, partially read papers and arXiv HTTP429/timeouts supply no negative evidence. No y=4, full-triangle, asymptotic or global-priority claim is included.