{"id":"e-d7adf0401ce51e02","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-e2f0f447a502f269","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":"306f089010964b55efde651b9941883f3e1960edef822e65f7e1f26191f52800","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":3736,"certificate_compressed_size":1271,"conclusion_name":"Triangle of second-order Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.","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-e2f0f447a502f269","claim":"interlacing"},{"node_id":"n-e2f0f447a502f269","claim":"real_rooted"},{"node_id":"n-e2f0f447a502f269","claim":"one_signed_coefficients"},{"node_id":"n-e2f0f447a502f269","claim":"hadamard_kernel"}]}