bibkey: mathar2012a108958 authors: R. J. Mathar year: 2012 title: “OEIS A108958, number of unordered pairs of distinct length-n binary words having the same number of 1’s: recurrence conjecture” doi: null url: https://oeis.org/A108958 claim: “The entry defines a(n) = Sum_{k=0..n} binomial(binomial(n,k),2) and records Mathar’s conjecture n(n-2)a(n) + 2(-3n^2+7n-3)a(n-1) + 4(n-1)(2n-3)a(n-2) = 0.” strata_touched:

• D5/S3/Combinatorics/ZeroQuantumPairsRecurrence license: citation-only triage: anchor

OEIS A108958

A108958 is “Number of unordered pairs of distinct length-n binary words having the same number of 1’s” (offset 1, data 0, 1, 6, 27, 110, 430, …). Its first formula line is

a(n) = Sum_{k=0..n} binomial(binomial(n, k), 2).

A comment by Stanislav Sykora (2012) reads

In coupled systems of n spin 1/2 particles (magnetic resonance) where the spin state of the i-th particle can be coded as 0 (Sz_i=-1/2) or 1 (Sz_i=+1/2), number of distinct (v<w) nontrivial (v!=w) zero-quantum transitions (v->w).

and the formula field contains

Conjecture: n*(n-2)a(n) +2(-3n^2+7n-3)a(n-1) +4(n-1)(2n-3) *a(n-2)=0. - R. J. Mathar, Apr 04 2012

Verified locator

• URL: https://oeis.org/A108958 (revision 77, last modified 2025-11-05, formula field; retrieved 2026-09-27 through the OEIS JSON interface).