{"id":"e-3183bd8d49ca6dec","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-daeeef1566d61733","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":"f9f8c5a072ec8e84a2c8deb15776f5fbf71a079d484394b7f6c3480c4f633afe","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":12517,"certificate_compressed_size":2202,"conclusion_name":"basis_map_A013609 of shift_x_plus_one of Triangle read by rows: T(n,k) = ((n+1)(n+2)/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-af416c91e2aef37c","role":"sequence","name":"shift_x_plus_one of Triangle read by rows: T(n,k) = ((n+1)(n+2)/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-daeeef1566d61733","claim":"one_signed_coefficients"},{"node_id":"n-daeeef1566d61733","claim":"hadamard_kernel"},{"node_id":"n-daeeef1566d61733","claim":"real_rooted"}]}