bibkey: curtz2011a189733
authors: Paul Curtz
year: 2011
title: "OEIS A189733, denominators of the first row of a rational difference array"
doi: null
url: https://oeis.org/A189733
claim: "a(n) = denominator(B(0,n)). Conjecture: a(6*n)=1, a(1+6*n)=1+3*n, a(2+6*n)=1+2*n, a(3+6*n)=2+3*n, a(4+6*n)=1, a(5+6*n)=6+6*n. a(n) = 2*a(n-6) - a(n-12)."
strata_touched:
  - D5/S1/Recurrence/Residue/OEISA189733DenominatorConjecture
license: citation-only
triage: anchor

OEIS A189733

Paul Curtz’s entry defines an array by its diagonal, first superdiagonal, and the recurrence of first differences. Its first-row reduced denominators are conjectured to obey six formulas indexed by the residue modulo six.

The source JSON at https://oeis.org/search?q=id%3AA189733&fmt=json, revision 36, carries the quoted conjecture. Its NAME line reads:

Denominator of B(0,n) where B(n,n)=0, B(n-1,n) = (-1)^(n+1)/n, and B(m,n) = B(m+1,n-1) + B(m,n-1), n >= 0, m >= 0, is an array of fractions.

The Maple and Mathematica definitions give boundary cases precedence. The upper-triangular part suffices for the first row. Setting T(m,k)=B(m,m+k) gives T(m,0)=0, T(m,1)=(-1)^m/(m+1), and T(m,k+2)=T(m,k+1)+T(m+1,k). The signs agree because m and m+2 have the same parity. The lower-triangular extension is not needed.

The exact universally quantified statement and source boundary are also recorded at https://github.com/the-omega-institute/trureturing/issues/25342.

Verified locator

• URL: https://oeis.org/A189733
• Source JSON: https://oeis.org/search?q=id%3AA189733&fmt=json, revision 36.