Metadata
bibkey: hanna2024a375449a375452
authors: Paul D. Hanna
year: 2024
title: "OEIS A375449 and A375452, diagonal derivative parity conjectures"
doi: null
url: https://oeis.org/A375449
claim: "For n > 1, the coefficients in each of the two diagonal derivative generating functions are odd iff n is of the form 2^k-2 or 2^k-1 for some k > 1."
strata_touched:
- D5/S1/Recurrence/Invariants/HannaDiagonalDerivativeParity
- D5/S1/Recurrence/Invariants/HannaDiagonalDerivativeParityA375452
license: citation-only
triage: anchor
Verified locator
DOI: null
Source: https://oeis.org/A375449
Sibling entry: https://oeis.org/A375452
Source statements
The unpaginated OEIS A375449 Name field states:
Expansion of g.f. A(x) satisfying [x^n] xA’(x) / (1 - nA(x)) = n^(n+1) for n >= 1.
The unpaginated OEIS A375452 Name field states:
Expansion of g.f. A(x) satisfying [x^n] xA’(x) / (1 - nA(x)) = n^n for n >= 1.
Each entry’s Comments field states:
Conjecture: for n > 1, a(n) is odd iff n is of the form 2^k-2 or 2^k-1 for some k > 1.
Both entries have offset one. The rational generating series has zero constant coefficient. Its formal derivative and unit-denominator inverse give the literal coefficient equation. The coefficient is identified with an integer before asserting oddness; indices and support exponents are natural numbers.
Mathematical scope
The diagonal recursion has an integral solution for every integer forcing sequence. When the forcing value at each positive index is congruent to that index modulo two, the reduced solution is T+T², where T has support at the positive Mersenne indices. This gives the odd and even support families in the quoted conjectures. The two forcing sequences are n^n and n^(n-1).
The source’s publication status and the completeness of the literature screen remain ASSUMED-UNVERIFIED beyond the quoted entries and the source screens in issues #18827 and #18826. No originality claim is made.