{"id":"e-ecef41f482d88baf","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-02d4e89e51dfb496","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":"b9376239f04293d6ebe0fce6d26019263405e276c4787984625946b4e1bd94ff","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":6879,"certificate_compressed_size":1393,"conclusion_name":"basis_map_A013611 of Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-eed4f744ce063213","role":"sequence","name":"Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.","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-02d4e89e51dfb496","claim":"real_rooted"},{"node_id":"n-02d4e89e51dfb496","claim":"one_signed_coefficients"},{"node_id":"n-02d4e89e51dfb496","claim":"hadamard_kernel"}]}