{"id":"e-deeee4aa3e096f87","operation":{"operation":"theorem_check","theorem":"liu_wang_three_term"},"operation_detail":"liu_wang_three_term","conclusion_id":"n-8a65e165240c40e0","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; Proposition 3.3 (the case B_n(x) = -(c_n x + d_n)^2)","lean_route":"sign","lean_declarations":["RealRooted.threeTerm_interlaces_of_eval_nonpos","RealRooted.threeTerm_interlaces","RealRooted.eval_nonpos_of_natDegree_le_two"],"certificate_hash":"85545cdd99cfe479591037681844e58bbf75377d2a13e43b5c3af8967209ff35","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":6258,"certificate_compressed_size":1905,"conclusion_name":"Triangular array of generalized Narayana numbers T(n,k) = 2*binomial(n+1,k)* binomial(n-2,k-1)/(n+1) for n >= 1 and 0 <= k <= n-1, read by rows.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[],"cited_by":[{"node_id":"n-8a65e165240c40e0","claim":"one_signed_coefficients"},{"node_id":"n-8a65e165240c40e0","claim":"hadamard_kernel"},{"node_id":"n-8a65e165240c40e0","claim":"interlacing"},{"node_id":"n-8a65e165240c40e0","claim":"real_rooted"}]}