{"id":"e-300346a71b22e19f","operation":{"operation":"theorem_check","theorem":"liu_wang_three_term"},"operation_detail":"liu_wang_three_term","conclusion_id":"n-c2eaf5212a4dec13","premise_count":0,"theorem":"L. L. Liu and Y. Wang, Adv. Appl. Math. 38 (2007) 542-560, Theorem 2.1 and Corollary 2.4 (A unified approach to polynomial sequences with only real zeros; the lag row as the interlacer, b_n = 0), weak interlacing of consecutive rows, with nonnegativity of the coefficients replaced, where a root region is certified, by an exact root-location induction; Proposition 3.3 (the case B_n(x) = -(c_n x + d_n)^2)","lean_route":"sign","lean_declarations":["RealRooted.threeTerm_interlaces_of_eval_nonpos","RealRooted.threeTerm_interlaces","RealRooted.eval_nonpos_of_natDegree_le_two"],"certificate_hash":"5e9bc3e1da777cd3fc7881c3f647080913c86788ec1aa1c906938130f4bb66d6","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":5426,"certificate_compressed_size":1586,"conclusion_name":"Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan triangle.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[],"cited_by":[{"node_id":"n-c2eaf5212a4dec13","claim":"real_rooted"},{"node_id":"n-c2eaf5212a4dec13","claim":"one_signed_coefficients"},{"node_id":"n-c2eaf5212a4dec13","claim":"hadamard_kernel"},{"node_id":"n-c2eaf5212a4dec13","claim":"interlacing"}]}