{"id":"e-49766d0fffe14f49","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-8cbd6010e0efad10","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":"7a98a3df96ba0341496f6f62c8c9d8e9d17c7a7be380340dda95c8baea6abc1b","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":13634,"certificate_compressed_size":2583,"conclusion_name":"basis_map_A013609 of Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).","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-c30cb06ad75c5e21","role":"sequence","name":"Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).","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-8cbd6010e0efad10","claim":"real_rooted"},{"node_id":"n-8cbd6010e0efad10","claim":"one_signed_coefficients"},{"node_id":"n-8cbd6010e0efad10","claim":"hadamard_kernel"},{"node_id":"n-8cbd6010e0efad10","claim":"interlacing"}]}