{"id":"e-ba7f4c6c74dd05c0","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-21854f4257effb01","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":"01f87e6ceed8a148e77e0f5407da3a7bbe20d024db74ce9db021329563186d61","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":11588,"certificate_compressed_size":2155,"conclusion_name":"Triangle read by rows: T(n,k) = binomial(k,n-k).","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-27bb5540b6efd4ae","role":"sequence","name":"basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k).","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"1"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 1]"}],"cited_by":[{"node_id":"n-21854f4257effb01","claim":"real_rooted"},{"node_id":"n-21854f4257effb01","claim":"one_signed_coefficients"},{"node_id":"n-21854f4257effb01","claim":"hadamard_kernel"}]}