{"id":"e-cc03321f42af72b5","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-c840f15d56a8a84f","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":"ffb2cbda75f43af12bc175cb2a12f97af20c97f113b3c25f67f2aaae849692d8","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":6889,"certificate_compressed_size":1404,"conclusion_name":"basis_map_A013610 of basis_map_A013611 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-98c21bd2a6c7c1d4","role":"sequence","name":"basis_map_A013611 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).","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-c840f15d56a8a84f","claim":"real_rooted"},{"node_id":"n-c840f15d56a8a84f","claim":"one_signed_coefficients"},{"node_id":"n-c840f15d56a8a84f","claim":"hadamard_kernel"}]}