{"id":"e-efe29139e2cf8aeb","operation":{"operation":"theorem_check","theorem":"liu_wang_three_term"},"operation_detail":"liu_wang_three_term","conclusion_id":"n-0a5e54eacdb24124","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":null,"lean_declarations":["RealRooted.MaWang.strictInterl_of_interlaces_evalCoeff_nonpos"],"certificate_hash":"fb5039a3e464e5ad7b269dbe7d22ede2219f5bee3ed364973aaa152d8ceee83c","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":6098,"certificate_compressed_size":1715,"conclusion_name":"Triangle read by rows. T(n, k) = (1/2) * binomial(2*(n - k + 1), n - k + 1) * binomial(2*n - k, k - 1) for n > 0, T(0, 0) = 1.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[],"cited_by":[{"node_id":"n-0a5e54eacdb24124","claim":"real_rooted"},{"node_id":"n-0a5e54eacdb24124","claim":"one_signed_coefficients"},{"node_id":"n-0a5e54eacdb24124","claim":"hadamard_kernel"}]}