{"id":"e-0bc3868d003d16cc","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-6587bc79a3c1f503","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":"771b5c4fd833e6ed16eeeb13bb941acd0c73c77b4e6a398ee524e48aa1ee4754","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":16457,"certificate_compressed_size":3228,"conclusion_name":"basis_map_A007318 of Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 3^(n-1)*binomial(n-2, k -1) otherwise.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":3,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-fbab854f2622405b","role":"sequence","name":"Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 3^(n-1)*binomial(n-2, k -1) otherwise.","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-6587bc79a3c1f503","claim":"real_rooted"},{"node_id":"n-6587bc79a3c1f503","claim":"one_signed_coefficients"},{"node_id":"n-6587bc79a3c1f503","claim":"hadamard_kernel"},{"node_id":"n-6587bc79a3c1f503","claim":"interlacing"}]}