{"id":"e-cae95ae48665cd8b","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-e0a0f9191d8a0512","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":"67f3fde21d5802ddde8076f001bf5578c3dcbb923e7cd135a869e86ebd8caf4f","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":13431,"certificate_compressed_size":2541,"conclusion_name":"basis_map_A007318 of Triangle read by rows: coefficients of the polynomials given by KummerU(-n, 1 - n - x, 3).","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-ff69dadb0811f5e6","role":"sequence","name":"Triangle read by rows: coefficients of the polynomials given by KummerU(-n, 1 - n - x, 3).","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-e0a0f9191d8a0512","claim":"one_signed_coefficients"},{"node_id":"n-e0a0f9191d8a0512","claim":"hadamard_kernel"},{"node_id":"n-e0a0f9191d8a0512","claim":"interlacing"},{"node_id":"n-e0a0f9191d8a0512","claim":"real_rooted"}]}