{"id":"e-d85043437d73496a","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-244a69427003c906","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":"5e6aef6941cdfbd21c8b02d36f591d70f2a3cebc56fa435a974754e13860ff40","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":15247,"certificate_compressed_size":2815,"conclusion_name":"shift_x_plus_one of derivative of Triangle of coefficients in expansion of (1+4x)^n.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-8d4f9cf28f37366a","role":"sequence","name":"derivative of Triangle of coefficients in expansion of (1+4x)^n.","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-244a69427003c906","claim":"one_signed_coefficients"},{"node_id":"n-244a69427003c906","claim":"hadamard_kernel"},{"node_id":"n-244a69427003c906","claim":"real_rooted"},{"node_id":"n-244a69427003c906","claim":"interlacing"}]}