{"id":"e-e0b14bd13b542826","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-07af2e6e3c9f0c77","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":"28e31c85f689af7e6e9e40bcd209bbbbeb6ae38650c253386ab0705e0a39a432","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":13652,"certificate_compressed_size":2542,"conclusion_name":"basis_map_A007318 of Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.","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-66e2c3448245b0b7","role":"sequence","name":"Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.","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-07af2e6e3c9f0c77","claim":"real_rooted"},{"node_id":"n-07af2e6e3c9f0c77","claim":"one_signed_coefficients"},{"node_id":"n-07af2e6e3c9f0c77","claim":"hadamard_kernel"},{"node_id":"n-07af2e6e3c9f0c77","claim":"interlacing"}]}