{"id":"e-ed772f252d15d000","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-b289b6e9a15ed6af","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":"b6e7da7459d19e02d79babc40071e6809e260b7992b935e8526cedeb8a068917","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":10659,"certificate_compressed_size":2341,"conclusion_name":"basis_map_A007318 of Triangle T(n, k) = 3*binomial(n,k) with T(0, 0) = 1, read by rows.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-7f71648c3ccd02d4","role":"sequence","name":"Triangle T(n, k) = 3*binomial(n,k) with T(0, 0) = 1, read by rows.","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-b289b6e9a15ed6af","claim":"real_rooted"},{"node_id":"n-b289b6e9a15ed6af","claim":"one_signed_coefficients"},{"node_id":"n-b289b6e9a15ed6af","claim":"hadamard_kernel"},{"node_id":"n-b289b6e9a15ed6af","claim":"interlacing"}]}