{"id":"e-92a0376fde876b5f","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-8006e68a699fe151","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":"bcb4b2f9663ec341e938561fdda665cc998ab6e52ba56cf437a9e2d9b38ae115","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":10665,"certificate_compressed_size":2347,"conclusion_name":"basis_map_A007318 of Triangle T(n,k), read by rows, given by (1,1,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":3,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-f2a2a3d7ea29bcbf","role":"sequence","name":"Triangle T(n,k), read by rows, given by (1,1,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}],"cited_by":[{"node_id":"n-8006e68a699fe151","claim":"real_rooted"},{"node_id":"n-8006e68a699fe151","claim":"one_signed_coefficients"},{"node_id":"n-8006e68a699fe151","claim":"hadamard_kernel"},{"node_id":"n-8006e68a699fe151","claim":"interlacing"}]}