{"id":"e-77ee5abd8546bf84","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-850ad8a1478de773","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":"cf8b7cbb5e06ee92bc29b22289a0cb411d18699f6a71dd3d174a47af4274d5e9","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":14502,"certificate_compressed_size":2879,"conclusion_name":"basis_map_A013609 of Triangle T(n,k) read by rows: associated Stirling numbers of first kind (n >= 2, 1 <= k <= floor(n/2)).","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-f13c6b611f4ce3bb","role":"sequence","name":"Triangle T(n,k) read by rows: associated Stirling numbers of first kind (n >= 2, 1 <= k <= floor(n/2)).","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-850ad8a1478de773","claim":"real_rooted"},{"node_id":"n-850ad8a1478de773","claim":"one_signed_coefficients"},{"node_id":"n-850ad8a1478de773","claim":"hadamard_kernel"},{"node_id":"n-850ad8a1478de773","claim":"interlacing"}]}