{"id":"e-06cd9a9d948af774","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-c2648604ab227c74","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":"3158f173bb1039a7cee92c38f997eaa70f6a022b38be2128ddf52d45613ea35c","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":13345,"certificate_compressed_size":2512,"conclusion_name":"basis_map_A007318 of Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-353fb5a029e4e669","role":"sequence","name":"Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.","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-c2648604ab227c74","claim":"real_rooted"},{"node_id":"n-c2648604ab227c74","claim":"one_signed_coefficients"},{"node_id":"n-c2648604ab227c74","claim":"hadamard_kernel"},{"node_id":"n-c2648604ab227c74","claim":"interlacing"}]}