{"id":"e-cb155a59ad1c4d66","operation":{"operation":"theorem_check","theorem":"liu_wang_three_term"},"operation_detail":"liu_wang_three_term","conclusion_id":"n-09153823987c7e65","premise_count":0,"theorem":"L. L. Liu and Y. Wang, Adv. Appl. Math. 38 (2007) 542-560, Theorem 2.1 and Corollary 2.4 (A unified approach to polynomial sequences with only real zeros; the lag row as the interlacer, b_n = 0), weak interlacing of consecutive rows, with nonnegativity of the coefficients replaced, where a root region is certified, by an exact root-location induction; Theorem 2.3 with the interlacers P'_(n-1) and P_(n-2) (Corollary 2.4 with b_n = V_n)","lean_route":null,"lean_declarations":[],"certificate_hash":"4e1561a0a684badce1eb199172b938bdcf6a82b4de8c4f6c5406575746689e1a","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":5115,"certificate_compressed_size":1487,"conclusion_name":"Triangle read by rows, related to number of permutations of [n] with 0 successions and k rises.","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-09153823987c7e65","claim":"real_rooted"},{"node_id":"n-09153823987c7e65","claim":"one_signed_coefficients"},{"node_id":"n-09153823987c7e65","claim":"hadamard_kernel"},{"node_id":"n-09153823987c7e65","claim":"interlacing"}]}