{"id":"e-46a7ecbf5694e808","operation":{"operation":"theorem_check","theorem":"liu_wang_three_term"},"operation_detail":"liu_wang_three_term","conclusion_id":"n-148a066ece673d8a","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":"3c276e818ca1a2d8b9eddee52513d5e01423aacef464cb77cfb9d3f9af561600","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":5597,"certificate_compressed_size":1627,"conclusion_name":"Regular triangle read by rows. T(n, k) = [[n, k]], where [[n, k]] are the second order Stirling cycle numbers (or second order reciprocal Stirling numbers). T(n, k) for 0 <= k <= n.","claims":[{"claim":"real_rooted","from_index":3,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":4,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[],"cited_by":[{"node_id":"n-148a066ece673d8a","claim":"one_signed_coefficients"},{"node_id":"n-148a066ece673d8a","claim":"hadamard_kernel"},{"node_id":"n-148a066ece673d8a","claim":"interlacing"},{"node_id":"n-148a066ece673d8a","claim":"real_rooted"}]}