{"id":"e-fcd19baf8f835cf0","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-392f49b496f79fbf","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":"d5d9576b47c874c4596b3ce911027452c4f2b3f9ea61cdc7b8376e6d8fded954","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":11671,"certificate_compressed_size":2194,"conclusion_name":"Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-500943bf802dcc95","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":"1"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 1]"}],"cited_by":[{"node_id":"n-392f49b496f79fbf","claim":"real_rooted"},{"node_id":"n-392f49b496f79fbf","claim":"one_signed_coefficients"},{"node_id":"n-392f49b496f79fbf","claim":"hadamard_kernel"}]}