{"id":"e-f7365f1dbad216ac","operation":{"operation":"parent_provers","route":"root_domain"},"operation_detail":"root_domain","conclusion_id":"n-fefc87d068bda822","premise_count":0,"theorem":"polytool parent provers, route root_domain: every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign","lean_route":null,"lean_declarations":[],"certificate_hash":"9782df96d451f899548870bcced51364d20a993bc158e2bfad5b8865faad20e1","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"parent_provers","certificate_raw_size":1180,"certificate_compressed_size":615,"conclusion_name":"Triangle read by rows, Stirling cycle numbers of order 3, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-1,k-1)+(n-1)^3*T(n-1,k), for n>=0 and 0<=k<=n.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[],"cited_by":[{"node_id":"n-fefc87d068bda822","claim":"real_rooted"},{"node_id":"n-fefc87d068bda822","claim":"one_signed_coefficients"},{"node_id":"n-fefc87d068bda822","claim":"hadamard_kernel"}]}