Sums of powers in the hybrid base {C, S, N}: factorization of the initial sequences, diagonals of the connection matrix and integrality

The sum σ_p(n) = Σ_{k=1}^{n} k^p admits expressions in terms of integer sequences (binomial coefficients, sums of consecutive binomial coefficients, and Narayana numbers). For odd p, σ_p contains a top-degree initial sequence D_p, and the coefficients of σ_p in the base {D_q} form a triangular matrix M, the connection matrix with the Faulhaber polynomials in u = σ_1. Two earlier conjectures of the author (Conjectures 3.1 and 3.2) propose the D_p explicitly. We take these expressions as a definition and prove, without further hypotheses on them: (i) the factorization of D_p in u, with triangular roots except for one linear root (Proposition 4.1 is an algebraic identity starting from the expression in Conjecture 3.2; Proposition 4.2 records the normalization in Conjecture 3.1); (ii) closed formulas for the second and third diagonals of M, and their relation to B_2 and B_4; (iii) an exact integrality criterion for the second diagonal: for p = 4K - 1 it is a half-integer if and only if the odd part m of K satisfies m ≥ 7 and m ≡ 3 (mod 4). Whether the D_p so defined are the initial sequences of σ_p is the content of the conjectures, and lies outside the scope of this work.