{"id":"e-e45b395d39c44382","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-2b8510f8f6f9afcd","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":"30a27c4e862239ac01812720f3307e9cf12f8208fe8be6a8abe623c24bbdb9e1","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":11707,"certificate_compressed_size":2117,"conclusion_name":"shift_x_plus_one of derivative of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-b60ac0f95ba69845","role":"sequence","name":"derivative of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.","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-2b8510f8f6f9afcd","claim":"one_signed_coefficients"},{"node_id":"n-2b8510f8f6f9afcd","claim":"hadamard_kernel"},{"node_id":"n-2b8510f8f6f9afcd","claim":"real_rooted"}]}