{"id":"e-a33c0ab38a356260","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-1b45c74901cc1e95","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":"da3e2c8f57b98a796053f5e39a4fa85a835b7d9555df6952d176ff8e93d195a6","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":7451,"certificate_compressed_size":1630,"conclusion_name":"basis_map_A007318 of Unsigned 2-Stirling numbers of the first kind.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-c237629cf5fdbb07","role":"sequence","name":"Unsigned 2-Stirling numbers of the first kind.","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-1b45c74901cc1e95","claim":"real_rooted"},{"node_id":"n-1b45c74901cc1e95","claim":"one_signed_coefficients"},{"node_id":"n-1b45c74901cc1e95","claim":"hadamard_kernel"}]}