{"id":"e-1ce099339e80292c","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-7b6a59413dc1b477","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":"9b06622db45e4fbded2a797edbea33b881af8092ad41d0e268cab2c521539acb","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":3719,"certificate_compressed_size":1247,"conclusion_name":"Triangle T(n,k), 0 <= k <= n, read by rows: given by [ 1, 0, 3, 0, 5, 0, 7, 0, 9, 0, ...] DELTA [ 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, ...] where DELTA is the operator defined in A084938.","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-7b6a59413dc1b477","claim":"real_rooted"},{"node_id":"n-7b6a59413dc1b477","claim":"one_signed_coefficients"},{"node_id":"n-7b6a59413dc1b477","claim":"hadamard_kernel"},{"node_id":"n-7b6a59413dc1b477","claim":"interlacing"}]}