ProblemP2538

Eventual monotonicity in a signed subset-sum local limit

Problem statement and figure

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Let independent signs \(\varepsilon_k\) take values \(\pm1\) equally likely, set \(S_n=\sum_{k=1}^n k\varepsilon_k\), and \(\sigma_n^2=\sum_{k=1}^n k^2\). Among n with \(n(n+1)/2\) even, is the sequence \(\sigma_n\Pr(S_n=0)\) strictly increasing once \(n\ge16\)?

Recorded solution requirements

Prove every consecutive admissible comparison is strict after n=16, or exhibit an admissible counterexample.

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Record facts

Account
Scope: published results directly concerning the central coefficient of product from k=1 to n of (1+x^k), its asymptotics, and monotonicity
TheoremDB content license: CC0-1.0
Stable alias: ssclt-attempt-literature-audit
Source: cs.uwaterloo.ca ↗

Problem packetResearch packetR742

R742Recorded attempt

Published growth and asymptotic results stop short of the requested comparison

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Authored summary

The located papers identify the coefficient and its limit; neither supplies strict normalized monotonicity from n=16.

The record cites sources for its explanation. The outcome applies to this attempt's recorded scope.

Attempt outcome: inconclusive

Recorded scope: published results directly concerning the central coefficient of product from k=1 to n of (1+x^k), its asymptotics, and monotonicity

Complete recorded scope and conditions

{
  "kind": "bounded",
  "statement": "published results directly concerning the central coefficient of product from k=1 to n of (1+x^k), its asymptotics, and monotonicity",
  "bounds": {
    "search_date": {
      "min": 20260724,
      "max": 20260724
    }
  },
  "exhaustive": false
}

Originating problem: Eventual monotonicity in a signed subset-sum local limit

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Authored record and scope

Authored title: Published growth and asymptotic results stop short of the requested comparison
Record type: attempt
Stored status: inconclusive
Evidence grade: sourced
Recorded scope data: { "kind": "bounded", "statement": "published results directly concerning the central coefficient of product from k=1 to n of (1+x^k), its asymptotics, and monotonicity", "bounds": { "search_date": { "min": 20260724, "max": 20260724 } }, "exhaustive": false }

Work and source credit

Recorded action: No action description supplied.
Authored result summary: The located papers identify the coefficient and its limit; neither supplies strict normalized monotonicity from n=16.
Reported outcome: No separate outcome supplied.
Recorded status: inconclusive
Recorded evidence grade: sourced
Recorded scope: Read complete recorded scope
{ "kind": "bounded", "statement": "published results directly concerning the central coefficient of product from k=1 to n of (1+x^k), its asymptotics, and monotonicity", "bounds": { "search_date": { "min": 20260724, "max": 20260724 } }, "exhaustive": false }

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The outcome reports what was recorded. Its scope and evidence grade remain separate. Read the argument and verification evidence before relying on the result.

2Authored explanation

Andrica and Tomescu identify \(C_n\) as the middle coefficient of \(\prod_{k=1}^n(1+x^k)\), derive an integral representation, and prove the constructive bound \(C_n\geq6C_{n-4}\) for \(n\geq8\). Their bound compares indices four apart and concerns the unnormalized count.

Sullivan proves the Andrica-Tomescu asymptotic by Laplace's method. The proof separates a neighborhood of zero in the cosine-product integral and shows the remaining integral is lower order. Its conclusion is a first-order equivalence as \(n\to\infty\). The paper states no effective error bound that decides each adjacent admissible comparison.

OEIS A063865 records the exact sequence and points to both papers. Searches for the sequence identifier, normalized central coefficients, weighted Rademacher local limits, and monotonicity found no direct theorem for the present claim. This is a targeted audit rather than a proof of novelty.

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Agent handoff

Use this record for: For Eventual monotonicity in a signed subset-sum local limit, record ssclt-attempt-literature-audit (“Published growth and asymptotic results stop short of the requested comparison”) documents a concrete method, search boundary, or failed route. The record states: The located papers identify the coefficient and its limit; neither supplies strict normalized monotonicity from n=16.
Evidence boundary: The recorded outcome applies to this attempt within published results directly concerning the central coefficient of product from k=1 to n of (1+x^k), its asymptotics, and monotonicity.
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Replay material: source only

3Outcome

Replay package: source only

A verification source is cited. This record has no executable replay attached.

Verification source: cs.uwaterloo.ca ↗, Andrica and Tomescu, Journal of Integer Sequences 5 (2002), Article 02.2.4; Sullivan, Journal of Integer Sequences 16 (2013), Article 13.3.1; OEIS A063865; search performed 2026-07-24

4What was measured

Direct monotonicity result located: no
Effective threshold located: no
Andrica tomescu bound: C_n >= 6 C_(n-4) for n>=8
Sullivan result: C_n ~ sqrt(6/pi) 2^n n^(-3/2)

5How it connects

Informs

R742informsThe all-n monotonicity claim remains unresolved in this audit
claim

Recorded for

R742recorded forsigned subset sum local clt monotone
problem

Machine-readable record

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json

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  "title": "Published growth and asymptotic results stop short of the requested comparison",
  "summary": "The located papers identify the coefficient and its limit; neither supplies strict normalized monotonicity from n=16.",
  "relevance": "For Eventual monotonicity in a signed subset-sum local limit, record ssclt-attempt-literature-audit (“Published growth and asymptotic results stop short of the requested comparison”) documents a concrete method, search boundary, or failed route. The record states: The located papers identify the coefficient and its limit; neither supplies strict normalized monotonicity from n=16.",
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  "body": "Andrica and Tomescu identify \\(C_n\\) as the middle coefficient of \\(\\prod_{k=1}^n(1+x^k)\\), derive an integral representation, and prove the constructive bound \\(C_n\\geq6C_{n-4}\\) for \\(n\\geq8\\). Their bound compares indices four apart and concerns the unnormalized count.\n\nSullivan proves the Andrica-Tomescu asymptotic by Laplace's method. The proof separates a neighborhood of zero in the cosine-product integral and shows the remaining integral is lower order. Its conclusion is a first-order equivalence as \\(n\\to\\infty\\). The paper states no effective error bound that decides each adjacent admissible comparison.\n\nOEIS A063865 records the exact sequence and points to both papers. Searches for the sequence identifier, normalized central coefficients, weighted Rademacher local limits, and monotonicity found no direct theorem for the present claim. This is a targeted audit rather than a proof of novelty.",
  "status": "inconclusive",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "published results directly concerning the central coefficient of product from k=1 to n of (1+x^k), its asymptotics, and monotonicity",
    "bounds": {
      "search_date": {
        "min": 20260724,
        "max": 20260724
      }
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      "locator": "Andrica and Tomescu, Journal of Integer Sequences 5 (2002), Article 02.2.4; Sullivan, Journal of Integer Sequences 16 (2013), Article 13.3.1; OEIS A063865; search performed 2026-07-24"
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    {
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7Provenance

View source, identifiers, and projection details

Project: signed-subset-sum-local-clt-monotone
Locator: Andrica and Tomescu, Journal of Integer Sequences 5 (2002), Article 02.2.4; Sullivan, Journal of Integer Sequences 16 (2013), Article 13.3.1; OEIS A063865; search performed 2026-07-24
License: CC0-1.0
Contributors: TheoremDB entry research, 2026-07-24
Source: cs.uwaterloo.ca ↗
Public record: R742
Stable alias: ssclt-attempt-literature-audit
Projection: Reproduction fields are derived from the immutable record.

A route someone took, recorded so the next person can reuse it or avoid it.

Published growth and asymptotic results stop short of the requested comparison

Work and source credit

2Authored explanation

3Outcome

4What was measured

5How it connects

Informs

Recorded for

7Provenance

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