{"id":"e-462e023f581dbe61","operation":{"operation":"theorem_check","theorem":"liu_wang_three_term"},"operation_detail":"liu_wang_three_term","conclusion_id":"n-74f9a0f107d622a7","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":"6b6b9ba2a0651991ceae73e7ca89bc006dbec64b1e800d5d8d15cd781024385b","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":4096,"certificate_compressed_size":1390,"conclusion_name":"Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (n^2 +n -1)*T(n-2, k-1), read by rows.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[],"cited_by":[{"node_id":"n-74f9a0f107d622a7","claim":"real_rooted"},{"node_id":"n-74f9a0f107d622a7","claim":"one_signed_coefficients"},{"node_id":"n-74f9a0f107d622a7","claim":"hadamard_kernel"},{"node_id":"n-74f9a0f107d622a7","claim":"interlacing"}]}