Shorter recurrences for the complementary and alternating parity sequences

Determine whether the sequence counting standard Young tableaux of shape (n,n,n) with even entry T(2,1), and the alternating sequence s(n)=sum_{r=1}^n(-1)^r f^{(n,n,n)/(r,1)}, satisfy recurrences shorter than those inherited from the order-two recurrence for a(n).

Background

The sequence a(n) counts tableaux with odd T(2,1), while its complement counts tableaux with even T(2,1). The paper also introduces an alternating sum of the first-descent classes, related to the parity split by 2a(n)=fn,n,n+s(n).

Neither of these two sequences was listed in the OEIS at the time of writing. Since both inherit recurrences from the recurrence for a(n), the unresolved issue is whether they possess recurrences of smaller order or otherwise shorter form.

References

Do they satisfy shorter recurrences than the ones they inherit from $L$?

— The first descent in a standard Young tableau of shape $(n,n,n)$  (2610.03038 - Niu, 2 Oct 2026) in Section 7, “Further questions”