{"id":"e-4e3482570d7ce009","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-a9e3a6c6145e13ca","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":"3ae5f39b5d26d8d3d8895c5071195725427ededd3f259a2f8f79ab681505f617","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":6885,"certificate_compressed_size":1402,"conclusion_name":"basis_map_A013610 of Triangle read by rows: T(n,k) is the number of binary words of length n having degree of asymmetry equal to k (n >= 0; 0 <= k <= n/2).","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-b6aca576a689e9f0","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of binary words of length n having degree of asymmetry equal to k (n >= 0; 0 <= k <= n/2).","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-a9e3a6c6145e13ca","claim":"real_rooted"},{"node_id":"n-a9e3a6c6145e13ca","claim":"one_signed_coefficients"},{"node_id":"n-a9e3a6c6145e13ca","claim":"hadamard_kernel"}]}