{"id":"e-262500dd933dbe95","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-910ba4510ca288b7","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":"8bfbb1a1ee6e56cc2e5fbd77492869b6e67d93f14fa45629306d561e0fb21cb4","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":10278,"certificate_compressed_size":2313,"conclusion_name":"basis_map_A013609 of basis_map_A013611 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(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-b4b87865f0ed6035","role":"sequence","name":"basis_map_A013611 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(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-910ba4510ca288b7","claim":"real_rooted"},{"node_id":"n-910ba4510ca288b7","claim":"one_signed_coefficients"},{"node_id":"n-910ba4510ca288b7","claim":"hadamard_kernel"},{"node_id":"n-910ba4510ca288b7","claim":"interlacing"}]}