{"id":"e-69f6deb0f18b06c0","operation":{"operation":"theorem_check","theorem":"liu_wang_three_term"},"operation_detail":"liu_wang_three_term","conclusion_id":"n-be392fc9b98fb771","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","lean_route":"sign","lean_declarations":["RealRooted.threeTerm_interlaces_of_eval_nonpos","RealRooted.threeTerm_interlaces","RealRooted.eval_nonpos_of_natDegree_le_two"],"certificate_hash":"ef3ed7a7562e194e5628d9fe1520a00123c59f1635eeb286a78fad0b96f617f5","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":4821,"certificate_compressed_size":1510,"conclusion_name":"Triangle read by rows: T(n,k) = C(n+k,n)*C(n,k)/(k+1), for n >= 0, k = 0..n.","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-be392fc9b98fb771","claim":"real_rooted"},{"node_id":"n-be392fc9b98fb771","claim":"one_signed_coefficients"},{"node_id":"n-be392fc9b98fb771","claim":"hadamard_kernel"},{"node_id":"n-be392fc9b98fb771","claim":"interlacing"}]}