{"id":"e-46d699555fa04487","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-90af30f849fc554d","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":"0c04a250e967a310388f2aa961bfcfed8c1660a4d0fa37a28385164fa393b0ca","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":3299,"certificate_compressed_size":1121,"conclusion_name":"Triangle read by rows: T(n,k) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = 8.","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-90af30f849fc554d","claim":"one_signed_coefficients"},{"node_id":"n-90af30f849fc554d","claim":"hadamard_kernel"},{"node_id":"n-90af30f849fc554d","claim":"interlacing"},{"node_id":"n-90af30f849fc554d","claim":"real_rooted"}]}