{"id":"e-5735495da102ae98","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-74e46c56b4bf9150","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":"d1554511ee91bac052595c0c48a002828c5f08ca756678d850a6ce02e8032943","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":12891,"certificate_compressed_size":2243,"conclusion_name":"basis_map_A013609 of Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-984320118b653f6b","role":"sequence","name":"Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).","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-74e46c56b4bf9150","claim":"real_rooted"},{"node_id":"n-74e46c56b4bf9150","claim":"one_signed_coefficients"},{"node_id":"n-74e46c56b4bf9150","claim":"hadamard_kernel"}]}