Metadata
bibkey: mathar2016a174193
authors: R. J. Mathar; OEIS Foundation Inc.
year: 2016
title: OEIS A174193 — an order-four recurrence for an algebraic generating series
doi: null
url: https://oeis.org/A174193
claim: Mathar conjectures the displayed order-four polynomial-coefficient recurrence for the coefficients of the specified algebraic generating function.
strata_touched:
- D5/S1/Recurrence/Algebraic/MatharA174193OrderFourRecurrence
license: citation-only
triage: anchor
Verified locator
Source: https://oeis.org/A174193
The unnumbered FORMULA section has no page numbers. Its text is:
G.f.: (1-8*x+13*x^2+24*x^3-48*x^4-(1-6*x+x^2+34*x^3-26*x^4-4*x^5)*sqrt(1-4*x))/
(2*x^2*(1 -x)*(1- 4*x)^2)
Conjecture: -(n+2)*(404489*n-1866368)*a(n) +(3147815*n^2-8680082*n-14098754)*a(n-1) +2*(-2336439*n^2+652022*n+14514849)*a(n-2) +8*(-1044770*n^2+10585023*n-20370553)*a(n-3) +8*(642982*n-2004177)*(2*n-9)*a(n-4)=0. - _R. J. Mathar_, Jun 14 2016
OEIS OFFSET is 0,3; the first coefficient has index zero. OEIS attributes
its content under CC BY-SA 4.0. This note cites the source and quotes the
mathematical statement.
Background and scope
Robert Brignall, Shalosh B. Ekhad, Rebecca Smith and Vincent Vatter,
Almost avoiding permutations, Discrete Mathematics 309 (2009), 6626–6631.
DOI: 10.1016/j.disc.2009.06.027.
Source: https://arxiv.org/abs/2007.15111v1.
The end of Section 2 gives the same generating function as an empirical
candidate for the counting series of Av(321)^{+1}. Mathar’s recurrence
is an OEIS conjecture and does not occur in that paper.
The formal statement concerns the displayed algebraic series over the
rationals: a root s has constant coefficient one and square 1-4X, and
the series A satisfies D*A=P-B*s. Existence is part of the statement.
The recurrence quantifies over all such A and every natural n ≥ 4.
Lagged indices use natural subtraction; the polynomial scalars are rational.
Identification with permutation counts and the paper’s Section 4 injection
and all-t questions remain open in this treatment.
Independent computation
The experiment is in the-omega-institute/trureturing-experiments, commit
b7f0518818771c2150a082e79fdfe6abffd8598e, at
docs/reports/mathar-a174193/check.py.
Its SHA-256 is 38ed870ab06543b0ba8dffcac775a263b844902ff5fbd34f81c0432f1335bfbb.
The reported command python3 docs/reports/mathar-a174193/check.py exits 0
with ALL_CHECKS_OK. It checks both polynomial certificates, 30 OEIS
coefficients, 197 recurrence residuals for 4 ≤ n ≤ 200, hook-length
values through 25, and deletion counts through 7. These finite checks
corroborate the source correspondence and do not prove the universal recurrence.
The literature screen is recorded in issue #24106. Completeness of that
screen and historical novelty are ASSUMED-UNVERIFIED; the unread publisher
version and later non-arXiv literature are not negative evidence.