{"id":"e-ea0255d0cce87571","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-a6f953758a1a61c0","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":"cd97bec20694bb985452107d7b6b2964d19be2b145489755d435312880808556","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":6860,"certificate_compressed_size":1385,"conclusion_name":"shift_x_plus_one of basis_map_A007318 of Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-c20d7dca3713c0eb","role":"sequence","name":"basis_map_A007318 of Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).","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-a6f953758a1a61c0","claim":"one_signed_coefficients"},{"node_id":"n-a6f953758a1a61c0","claim":"hadamard_kernel"},{"node_id":"n-a6f953758a1a61c0","claim":"real_rooted"}]}