{"id":"e-26ea87317172461f","operation":{"map":"A130595","operation":"basis_map"},"operation_detail":"A130595","conclusion_id":"n-6c0f994846d774ff","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":"917f790814f2ea361af9f057a2d53249b5611a9e1572a1952672031711b0732f","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":6160,"certificate_compressed_size":1365,"conclusion_name":"Triangle read by rows: lower triangular matrix which is inverse to Pascal's triangle (A007318) regarded as a lower triangular matrix.","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-1bf99ad40f056d33","role":"sequence","name":"basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}],"cited_by":[{"node_id":"n-6c0f994846d774ff","claim":"real_rooted"}]}