{"id":"e-09aef78f63ec9a62","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-de5746895014fee3","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":"a89043961831fd9eb15ef297396d45ba77ef1de20535d0143ff4ffaf882e8b7a","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":3267,"certificate_compressed_size":1122,"conclusion_name":"Triangle read by rows, T(n, k) = 4^k*S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.","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-de5746895014fee3","claim":"real_rooted"},{"node_id":"n-de5746895014fee3","claim":"one_signed_coefficients"},{"node_id":"n-de5746895014fee3","claim":"hadamard_kernel"},{"node_id":"n-de5746895014fee3","claim":"interlacing"}]}