{"id":"e-400df3955695c308","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-431d0fbb7561f8e5","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":"cb6db20df816d43eed417b00098f2b5b3cc1831a750c8374066b98afaee604e9","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":12556,"certificate_compressed_size":2207,"conclusion_name":"basis_map_A013611 of Triangle read by rows: T(n,k) = (k+1)^2*binomial(n,k) (0 <= k <= n).","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-e7d0ffa461d9dbc4","role":"sequence","name":"Triangle read by rows: T(n,k) = (k+1)^2*binomial(n,k) (0 <= k <= n).","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-431d0fbb7561f8e5","claim":"real_rooted"},{"node_id":"n-431d0fbb7561f8e5","claim":"one_signed_coefficients"},{"node_id":"n-431d0fbb7561f8e5","claim":"hadamard_kernel"}]}