{"id":"e-f1f0bc7d532f0ef7","operation":{"operation":"theorem_check","theorem":"liu_wang_three_term"},"operation_detail":"liu_wang_three_term","conclusion_id":"n-71548025909043e7","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_nonnegCoeffs","RealRooted.threeTerm_interlaces"],"certificate_hash":"689ffd3aaa3a62d5f07495df89a1663e0fcbae7311059c4602109111ce63be04","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":4929,"certificate_compressed_size":1545,"conclusion_name":"Triangle T(n,k,p,q,j) = T(n-1,k,p,q,j) + T(n-1,k-1,p,q,j) + (p*j+q)*prime(j)*T(n-2,k-1,p,q,j) with T(2,k,p,q,j) = prime(j) and (p,q,j) = (0,1,2), read by rows.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[],"cited_by":[{"node_id":"n-71548025909043e7","claim":"real_rooted"},{"node_id":"n-71548025909043e7","claim":"one_signed_coefficients"},{"node_id":"n-71548025909043e7","claim":"hadamard_kernel"},{"node_id":"n-71548025909043e7","claim":"interlacing"}]}