{"id":"e-b33b68f249378f3c","operation":{"operation":"theorem_check","theorem":"liu_wang_three_term"},"operation_detail":"liu_wang_three_term","conclusion_id":"n-51a2369b25b8e3ba","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":"window","lean_declarations":["RealRooted.threeTerm_interlaces_of_roots_le"],"certificate_hash":"1116372759cf8124f6d3fd43f30f71891b7ce1589e5a18ba2f77e5c0ded5ec7f","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":3576,"certificate_compressed_size":1210,"conclusion_name":"shift_x_plus_one of Triangle read by rows: T(n,k) = binomial(k,n-k).","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[],"cited_by":[{"node_id":"n-51a2369b25b8e3ba","claim":"interlacing"},{"node_id":"n-51a2369b25b8e3ba","claim":"real_rooted"},{"node_id":"n-51a2369b25b8e3ba","claim":"one_signed_coefficients"},{"node_id":"n-51a2369b25b8e3ba","claim":"hadamard_kernel"}]}