OEIS A391810: formula (5), COMMENTS (C.1)--(C.10), and prime-index vanishing

We prove formula (5), COMMENTS (C.1)--(C.10), and the zero clause of COMMENT (C.11) of Paul D. Hanna's OEIS entry A391810, revision 39. For n>=1, the original bilateral Laurent coefficients form a finite row polynomial P_n(y)=-1+sum(h=1..n)y^h(1-y^h)^(n-h). Evaluating it at -1 gives the negative alternating row sum 2 round(2^n/3)=2J_n. The same polynomial proves formulas (2), (3), (4), and (6). Its divisor-sum coefficient expression proves the ten individual-term formulas and T(n,q)=0 for every prime q>n. The further first-zero assertion of (C.11) remains open in this note.