Binomial-Sum Divisibility, Integral Fourth Roots, and a Legendre Coefficient Identity

Consider a double binomial sum whose values at a fixed parameter form OEIS A098275 and whose leading parameter coefficients form OEIS A274786. We give self-contained proofs of divisibility by n + 1 and of integrality of a formal fourth root for the first sequence. A Vandermonde transformation supplies the former; a binary-digit divisibility estimate together with a finite, fully justified Laurent-polynomial congruence reduction supplies the latter. A separate Legendre-polynomial coefficient identity is obtained from the classical Clausen square formula and an explicit coefficient recurrence. The two sequences arise from a common polynomial family, but the three statements have distinct mathematical proofs. The previously published treatment of the two A098275 conjectures and the original OEIS formula contributors are acknowledged. Numerical verification against the available full b-file snapshots is supplementary and is not used in place of an infinite proof. Licensing note: The manuscript is licensed under CC BY 4.0. The OEIS b-file snapshots included in the supplementary archive retain their original CC BY-SA 4.0 license and attribution. The independently authored verification code is made available under CC BY 4.0.