{"id":"e-5f6d53acefa22cfd","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-daa3b3e1ead00edc","premise_count":0,"theorem":"L. L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Appl. Math. 38 (2007) 542-560, Theorem 2.1 with g = f' (as in their Corollary 2.4), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","lean_route":null,"lean_declarations":[],"certificate_hash":"9fdb208727b35c2cca088e98ffa0cdf8ef4a5beb5f9efee6de0bacc21131beaf","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":3299,"certificate_compressed_size":1130,"conclusion_name":"Triangle read by rows, coefficients of the generalized Eulerian polynomials A_{n, 3}(x) in descending order.","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[],"cited_by":[{"node_id":"n-daa3b3e1ead00edc","claim":"real_rooted"},{"node_id":"n-daa3b3e1ead00edc","claim":"one_signed_coefficients"},{"node_id":"n-daa3b3e1ead00edc","claim":"hadamard_kernel"},{"node_id":"n-daa3b3e1ead00edc","claim":"interlacing"}]}