{"id":"e-770b6439e58a100f","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-f8ceb867f43c1485","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":"3e7f4a8fbc1b8e8ed98a02438be0925b43c4cde86ff3667657dfba3e080e8c12","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":6879,"certificate_compressed_size":1395,"conclusion_name":"basis_map_A013611 of Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-336459bdce627a87","role":"sequence","name":"Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 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-f8ceb867f43c1485","claim":"real_rooted"},{"node_id":"n-f8ceb867f43c1485","claim":"one_signed_coefficients"},{"node_id":"n-f8ceb867f43c1485","claim":"hadamard_kernel"}]}