{"id":"e-11b86d48c22b376e","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-3d0ce3ba50a1aaca","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":"f852419f3c1ebac5325eeb32a960d9604ecba073584f04dbe9b0c58cdf7d7f75","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":13769,"certificate_compressed_size":2618,"conclusion_name":"basis_map_A013609 of One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.","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-2245b35ec7cd4857","role":"sequence","name":"One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.","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-3d0ce3ba50a1aaca","claim":"interlacing"},{"node_id":"n-3d0ce3ba50a1aaca","claim":"real_rooted"},{"node_id":"n-3d0ce3ba50a1aaca","claim":"one_signed_coefficients"},{"node_id":"n-3d0ce3ba50a1aaca","claim":"hadamard_kernel"}]}