{"id":"e-44764fae5f91faca","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-d9801c5264eca57f","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":"577adccbfc1c30052c904324b94d38cabe4bffde0b9a6a990d59e78f7693d2e1","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":14967,"certificate_compressed_size":3004,"conclusion_name":"shift_x_plus_one of basis_map_A013610 of Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one version of) Fibonacci polynomials.","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-054b8a71941f3c19","role":"sequence","name":"basis_map_A013610 of Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one version of) Fibonacci polynomials.","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-d9801c5264eca57f","claim":"one_signed_coefficients"},{"node_id":"n-d9801c5264eca57f","claim":"hadamard_kernel"},{"node_id":"n-d9801c5264eca57f","claim":"real_rooted"},{"node_id":"n-d9801c5264eca57f","claim":"interlacing"}]}