{"id":"e-ab2b07f2d97ec943","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-914700b1060feeb2","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":"79d02c5a36360f9810d7c5869af7a30036c37439780b9a4dd4c5504b9f3b1457","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":2747,"certificate_compressed_size":1034,"conclusion_name":"Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirling numbers of the first kind (n >= 2; 1 <= k <= n-1; s(n,j) = A008275(n,j)).","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-914700b1060feeb2","claim":"real_rooted"},{"node_id":"n-914700b1060feeb2","claim":"one_signed_coefficients"},{"node_id":"n-914700b1060feeb2","claim":"hadamard_kernel"},{"node_id":"n-914700b1060feeb2","claim":"interlacing"}]}