Metadata

bibkey: bala2023quarticparity
authors: Peter Bala
year: 2023
title: "OEIS A361034 and A361035: parity of quartic factorial quotients"
doi: null
url: https://oeis.org/A361034
claim: "Conjecture: a(n) is odd iff n = 2^k - 2 for some k >= 1."
strata_touched:
  - D5/S3/Arith/FactorialRatio/BalaA361034Parity
  - D5/S3/Arith/FactorialRatio/BalaA361035Parity
license: citation-only
triage: anchor

Quartic factorial quotient parity

Verified locator

Source: https://oeis.org/A361034

Companion source: https://oeis.org/A361035 The source entries are unpaginated. The quotations are from their NAME and COMMENTS fields in the OEIS data mirror oeis/oeisdata at e7d0073e.

Source statements

A361034

a(n) = 2520*(4n)!/(n!(n+2)!^3).

Conjecture: a(n) is odd iff n = 2^k - 2 for some k >= 1.

A361035

a(n) = 9979200 * (4n)!/(n!(n+3)!^3).

Conjecture: a(n) is odd iff n = 2^k - 3 for some k >= 2.

The natural index includes zero. Natural-number division represents the source quotient because its denominator divides its numerator. The exponent lower bounds ensure that the natural subtractions are exact.

Scope

The two rows have coefficients 2520 and 9979200 and shifts 2 and 3. Their common proof compares factorial prime exponents and uses the binary digit sum to determine parity. No conclusion about every row of A361032 is asserted. Literature completeness is ASSUMED-UNVERIFIED beyond the sources screened in issues #18606 and #18607.

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