{"id":"e-acbcb57301a85aa1","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-5d17ea896d70e4da","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":"fe2743d217c244ab8c103d069c5d47621fc9a9186f95334fde45e3b1f605036b","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":10523,"certificate_compressed_size":2327,"conclusion_name":"shift_x_plus_one of basis_map_A013611 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-77621b0cfa8249dd","role":"sequence","name":"basis_map_A013611 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= 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-5d17ea896d70e4da","claim":"one_signed_coefficients"},{"node_id":"n-5d17ea896d70e4da","claim":"hadamard_kernel"},{"node_id":"n-5d17ea896d70e4da","claim":"interlacing"},{"node_id":"n-5d17ea896d70e4da","claim":"real_rooted"}]}