{"id":"e-69217c00637554db","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-ec2cafb410ca50bb","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":"7555a5a7a5f483d018ad1cdc16b1dc98c4874484dc1e1f31ae10b5e88552aa30","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":14629,"certificate_compressed_size":2783,"conclusion_name":"basis_map_A007318 of Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + j*prime(j)*T(n-2, k-1) with j=3, read by rows.","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-27c21e0d08511c1c","role":"sequence","name":"Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + j*prime(j)*T(n-2, k-1) with j=3, 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-ec2cafb410ca50bb","claim":"real_rooted"},{"node_id":"n-ec2cafb410ca50bb","claim":"one_signed_coefficients"},{"node_id":"n-ec2cafb410ca50bb","claim":"hadamard_kernel"},{"node_id":"n-ec2cafb410ca50bb","claim":"interlacing"}]}