{"id":"e-b4e5a61cd5085168","operation":{"operation":"theorem_check","theorem":"liu_wang_three_term"},"operation_detail":"liu_wang_three_term","conclusion_id":"n-e48eea24800097e1","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":"075ed997a434087ca2798ca226bbceefddeba3aca3840eaaa866bcca218a64d4","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":5402,"certificate_compressed_size":1623,"conclusion_name":"Triangle read by rows: T(n,k) = binomial(n-2, k-1) + n*binomial(n-1, k-1), 1 <= k <= n, starting with T(1, 1) = 1.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}}],"premises":[],"cited_by":[{"node_id":"n-e48eea24800097e1","claim":"real_rooted"},{"node_id":"n-e48eea24800097e1","claim":"one_signed_coefficients"},{"node_id":"n-e48eea24800097e1","claim":"hadamard_kernel"}]}