{"id":"e-5b63109a4e23ed59","operation":{"operation":"theorem_check","theorem":"liu_wang_three_term"},"operation_detail":"liu_wang_three_term","conclusion_id":"n-353fb5a029e4e669","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","lean_route":"sign","lean_declarations":["RealRooted.threeTerm_interlaces_of_eval_nonpos","RealRooted.threeTerm_interlaces","RealRooted.eval_nonpos_of_natDegree_le_two"],"certificate_hash":"117b511fde292d5fbf6d4554048df906c552a45a85c70062bb56eda303b0f22d","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":4177,"certificate_compressed_size":1392,"conclusion_name":"Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.","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-353fb5a029e4e669","claim":"interlacing"},{"node_id":"n-353fb5a029e4e669","claim":"real_rooted"},{"node_id":"n-353fb5a029e4e669","claim":"one_signed_coefficients"},{"node_id":"n-353fb5a029e4e669","claim":"hadamard_kernel"}]}