{"id":"e-5dbc61b0bf2da0f7","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-d1857659a362b7dc","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":"cd8fd7a6ca82eb46691575e8d10ab652b57fc092ac0263e14f23407f809a684c","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":14963,"certificate_compressed_size":3011,"conclusion_name":"basis_map_A013609 of Triangle T(n,k), read by rows, given by (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.","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-72cbbb2f152c8f9c","role":"sequence","name":"Triangle T(n,k), read by rows, given by (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.","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-d1857659a362b7dc","claim":"real_rooted"},{"node_id":"n-d1857659a362b7dc","claim":"one_signed_coefficients"},{"node_id":"n-d1857659a362b7dc","claim":"hadamard_kernel"},{"node_id":"n-d1857659a362b7dc","claim":"interlacing"}]}