{"id":"e-169e30e3ff1aab79","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-2b8ae16f62b5e385","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":"e6075f3672e47af89a926906d607cdb1481569dbc9cf58d540bc887c70417fd8","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":13782,"certificate_compressed_size":2633,"conclusion_name":"basis_map_A013609 of 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-194cb2d73cc003c5","role":"sequence","name":"basis_map_A013610 of 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-2b8ae16f62b5e385","claim":"real_rooted"},{"node_id":"n-2b8ae16f62b5e385","claim":"one_signed_coefficients"},{"node_id":"n-2b8ae16f62b5e385","claim":"hadamard_kernel"},{"node_id":"n-2b8ae16f62b5e385","claim":"interlacing"}]}