{"id":"e-28a022daf4fe1fd9","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-1916a93c76107e9a","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":"10d53b2c16a989eaa9418a56211eead350810bb0338d683c96b3444c9105beaf","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":10244,"certificate_compressed_size":2288,"conclusion_name":"shift_x_plus_one of basis_map_A013609 of Triangle read by rows: row n consists of n zeros followed by n!.","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-17166a4ad91393c8","role":"sequence","name":"basis_map_A007318 of Triangle T(n, k) read by rows, T(n, k) = n!*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-1916a93c76107e9a","claim":"one_signed_coefficients"},{"node_id":"n-1916a93c76107e9a","claim":"hadamard_kernel"},{"node_id":"n-1916a93c76107e9a","claim":"interlacing"},{"node_id":"n-1916a93c76107e9a","claim":"real_rooted"}]}