{"id":"e-0b9b27d3516a8ce8","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-cb3338f436ef0131","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":"f86c99403a87383b4224e13493e97db153d2bcb5e10a1cfc4fe79155c188d87b","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":12294,"certificate_compressed_size":2365,"conclusion_name":"basis_map_A007318 of Triangle T(n,k) = 3*T(n-1,k) +  T(n-3,k-1) for k = 0..floor(n/3) with T(0,0) = 1 and T(n,k) = 0 for n or k < 0, read by rows.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-a5b33928f20c14a2","role":"sequence","name":"Triangle T(n,k) = 3*T(n-1,k) +  T(n-3,k-1) for k = 0..floor(n/3) with T(0,0) = 1 and T(n,k) = 0 for n or k < 0, read by rows.","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-cb3338f436ef0131","claim":"real_rooted"},{"node_id":"n-cb3338f436ef0131","claim":"one_signed_coefficients"},{"node_id":"n-cb3338f436ef0131","claim":"hadamard_kernel"}]}