{"id":"e-926e24a16f3abf3d","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-7f05fd9e9c6948d8","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":"c3e63e94a7fe716962ee6c62122cfc0ad3dfa44ddc982dbfd05293020f621fe6","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":6855,"certificate_compressed_size":1388,"conclusion_name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-8d88ef9de74b305f","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.","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-7f05fd9e9c6948d8","claim":"real_rooted"},{"node_id":"n-7f05fd9e9c6948d8","claim":"one_signed_coefficients"},{"node_id":"n-7f05fd9e9c6948d8","claim":"hadamard_kernel"}]}