{"id":"e-9ef185a97bf0710a","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-016d7724d51f6416","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":"81e4b61246b8b8763eeb97d472f24eea28141492e2f9f774260da1e4f204653f","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":14417,"certificate_compressed_size":2792,"conclusion_name":"Riordan array (1, 3+3x).","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-c7f2f3338041a8d8","role":"sequence","name":"basis-map coefficients of Riordan array (1, 3+3x).","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"1"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 1]"}],"cited_by":[{"node_id":"n-016d7724d51f6416","claim":"one_signed_coefficients"},{"node_id":"n-016d7724d51f6416","claim":"hadamard_kernel"},{"node_id":"n-016d7724d51f6416","claim":"real_rooted"},{"node_id":"n-016d7724d51f6416","claim":"interlacing"}]}