{"id":"e-367cc697b56efa4f","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-49350b617f7163d1","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":"8272f6bda3b4616b9fbe9bc16846f107d1155f2136a17aad921e9d6b67dcaa79","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":13217,"certificate_compressed_size":2516,"conclusion_name":"basis_map_A013610 of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).","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-dd77bf712b1df996","role":"sequence","name":"Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).","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-49350b617f7163d1","claim":"real_rooted"},{"node_id":"n-49350b617f7163d1","claim":"one_signed_coefficients"},{"node_id":"n-49350b617f7163d1","claim":"hadamard_kernel"},{"node_id":"n-49350b617f7163d1","claim":"interlacing"}]}