{"id":"e-f9693f57c5cb4021","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-194cb2d73cc003c5","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"93fd3f85fb374857769b11d36da5c30905e3a48aaaa4c485e4b6a56f131e3eec","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":13634,"certificate_compressed_size":2590,"conclusion_name":"basis_map_A013610 of Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-ba8de4e02acfb6ca","role":"sequence","name":"Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}],"cited_by":[{"node_id":"n-194cb2d73cc003c5","claim":"real_rooted"},{"node_id":"n-194cb2d73cc003c5","claim":"one_signed_coefficients"},{"node_id":"n-194cb2d73cc003c5","claim":"hadamard_kernel"},{"node_id":"n-194cb2d73cc003c5","claim":"interlacing"}]}