{"id":"e-0f37534727a0e0cd","operation":{"operation":"theorem_check","theorem":"three_term_recurrence"},"operation_detail":"three_term_recurrence","conclusion_id":"n-8ce61d302cec295b","premise_count":0,"theorem":"three-term recurrence Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2) with a_n != 0 and beta_n = c_n / (a_n a_(n-1)) > 0: simple real roots and strict interlacing of consecutive rows (zero separation for orthogonal polynomials: G. Szeg\u0151, Orthogonal Polynomials, \u00a73.3; Favard, C. R. Acad. Sci. Paris 200 (1935)); root regions by a positive-ratio (chain-sequence) induction","lean_route":null,"lean_declarations":[],"certificate_hash":"514bbbcb85dd01ff00aa143d64224bddd7c268e3d970054e7465eb2c44aac299","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":5259,"certificate_compressed_size":1477,"conclusion_name":"Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k double rises (0 <= k <= n-1).","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"strict","orientation":"previous_interlaces_next"}],"premises":[],"cited_by":[{"node_id":"n-8ce61d302cec295b","claim":"real_rooted"},{"node_id":"n-8ce61d302cec295b","claim":"one_signed_coefficients"},{"node_id":"n-8ce61d302cec295b","claim":"hadamard_kernel"},{"node_id":"n-8ce61d302cec295b","claim":"interlacing"}]}