{"id":"e-d9277c75939e8d2f","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-4155b181fd90d601","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":"b20ed9c8654c4de7e0c4ce2e232ecf0b93c40c74d06817d6a11d4f56b039e3f9","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":3299,"certificate_compressed_size":1135,"conclusion_name":"Triangle read by rows: T(n,k) = t(n-k, k), where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(n) = 5*n + 3.","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-4155b181fd90d601","claim":"real_rooted"},{"node_id":"n-4155b181fd90d601","claim":"one_signed_coefficients"},{"node_id":"n-4155b181fd90d601","claim":"hadamard_kernel"},{"node_id":"n-4155b181fd90d601","claim":"interlacing"}]}