{"id":"e-f1d68648f9912a0f","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-ba85471ec6672bd6","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":"ff295f7b77a1e5a7346efdccbbb61979fe6258b8c9b72c6a1f4919be0dafb606","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":6897,"certificate_compressed_size":1400,"conclusion_name":"shift_x_plus_one of basis_map_A007318 of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-39578f69b35e8349","role":"sequence","name":"basis_map_A007318 of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.","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-ba85471ec6672bd6","claim":"hadamard_kernel"},{"node_id":"n-ba85471ec6672bd6","claim":"real_rooted"},{"node_id":"n-ba85471ec6672bd6","claim":"one_signed_coefficients"}]}