{"id":"e-419a6747cb084155","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-4d60c3dd8627c01d","premise_count":0,"theorem":"L. L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Appl. Math. 38 (2007) 542-560, Theorem 2.1 with g = f' (as in their Corollary 2.4), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","lean_route":null,"lean_declarations":[],"certificate_hash":"addfe6bafcc78af13d1308a4b02f1b8c882d8713f898529a6cf371ea40c5a225","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":2735,"certificate_compressed_size":1033,"conclusion_name":"Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first kind in reverse order).","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-4d60c3dd8627c01d","claim":"real_rooted"},{"node_id":"n-4d60c3dd8627c01d","claim":"one_signed_coefficients"},{"node_id":"n-4d60c3dd8627c01d","claim":"hadamard_kernel"},{"node_id":"n-4d60c3dd8627c01d","claim":"interlacing"}]}