bibkey: oeis2024a329369 authors: OEIS Foundation Inc.; Mikhail Kurkov year: 2024 title: OEIS A329369 sixth conjectured sum over the A373183 rows doi: null url: https://oeis.org/A329369 claim: The sixth conjectured sum is printed with the index q unrestricted, and that reading is false. strata_touched:
- D5/S1/Recurrence/Parity/DyadicPowerRowClosedForm license: citation-only triage: anchor
OEIS A329369 sixth conjectured sum over the A373183 rows
Directly inspected on September 8, 2026 through the plain-text interface. The
title is “Number of permutations of {1,2,…,m} with excedance set constructed
by taking m-i (0 < i < m) if b(i-1) = 1 where b(k)b(k-1)…b(1)b(0)
(0 <= k < m-1) is the binary expansion of n.” The offset is 0,3 and Mikhail
Kurkov created the entry on November 12, 2019. The internal revision line at
inspection was %I A329369 #239 Jul 01 2026 05:33:40.
The formula block attributed to Mikhail Kurkov, June 05 2024, reads verbatim:
“Conjecture 6: a(2^m*n + q) = Sum_{i=A001511(n+1)..A000120(n)+1} A373183(n, i)a(2^m(2^(i-1)-1) + q) for n >= 0, m >= 0, q >= 0. Note that this formula is recursive for n != 2^k - 1. Also, it is not related to R. Ehrenborg’s and E. Steingrimsson’s work.”
The same assertion is printed a second time as A373183 Conjecture 2, in the
entry whose revision line read %I A373183 #42 Jul 08 2026 17:47:51. The two
printings are one assertion, not two.
The two defining recurrences printed in the same block are “a(2n+1) = a(n) for n >= 0” and “a(2n) = a(n) + a(n - 2^f(n)) + a(2n - 2^f(n)) for n > 0 with a(0) = 1 where f(n) = A007814(n)”. The listed initial terms are 1, 1, 3, 1, 7, 3, 7, 1, 15, 7, 17, 3, 31.
A373183 is described as “Irregular triangle T(n, k), n >= 0, k > 0, read by rows with row polynomials R(n, x) such that R(2n+1, x) = xR(n, x) for n >= 0, R(2n, x) = x(R(n, x+1) - R(n, x)) for n > 0 with R(0, x) = x”, and its comment states that row n has length A000120(n) + 1.
Verified locator
- https://oeis.org/A329369
- https://oeis.org/A373183
- https://oeis.org/search?q=id:A329369&fmt=text: title, offset, author, term list, the two defining recurrences, and the June 05 2024 formula block.
- https://oeis.org/search?q=id:A373183&fmt=text: the row polynomial definition, the row length comment, and the initial rows.
Scope of the formal work
The formal module reads the two defining recurrences as the evaluation at one of the A373183 row polynomials, and proves that reading from the row recursion itself rather than assuming it. It then computes the rows indexed by a positive power of two in closed form and evaluates both sides of the printed sum at m equal to zero and q equal to one.
What is and is not settled
The printed reading, in which q ranges over all natural numbers, is false. A direct sweep of the two defining recurrences over indices below 2^18, with n below 256 and m at most ten, found 2781 index triples where the two sides differ, and every one of them has q at least 2^m. The same sweep restricted to q below 2^m found no differing triple. So the assertion under that digit restriction is untouched here and remains open; the exponent m and the residue q form a base 2^m decomposition, which is the reading the restriction records.
The sweep is arithmetic evidence about a bounded index window, not a proof of the restricted assertion.
ASSUMED-UNVERIFIED
Whether the unrestricted range in the printed text is an oversight rather than the intended assertion is not established here; only the printed text is cited. No search for a published correction or for prior notice of this range was performed, so priority is not claimed.