{"id":"e-ced4e939f60ab674","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-606569787e0702d3","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":"b8a2a7dd67276151ed2d68e07c84bb02b67dd10f8f84d28386474207a60cf162","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":6887,"certificate_compressed_size":1397,"conclusion_name":"shift_x_plus_one of basis_map_A013611 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-7de8fc70cf185364","role":"sequence","name":"basis_map_A013611 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.","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-606569787e0702d3","claim":"one_signed_coefficients"},{"node_id":"n-606569787e0702d3","claim":"hadamard_kernel"},{"node_id":"n-606569787e0702d3","claim":"real_rooted"}]}