{"id":"e-0566c972c7be0ca0","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-2e0a8214849adcd6","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":"736c23a88ba07e2d84138264ecd10152ceb482a5638ba2703d34a6b6cef115f9","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":6913,"certificate_compressed_size":1410,"conclusion_name":"basis_map_A007318 of Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-26633b980c3508e8","role":"sequence","name":"Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}],"cited_by":[{"node_id":"n-2e0a8214849adcd6","claim":"real_rooted"},{"node_id":"n-2e0a8214849adcd6","claim":"one_signed_coefficients"},{"node_id":"n-2e0a8214849adcd6","claim":"hadamard_kernel"}]}