{"id":"e-e548dab430fb29bc","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-d28ad9841fc73317","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":"7fe9f55f95f4a78d283ca403b9e918c34c2ecf781e9b42c07df2c980607363e3","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":6855,"certificate_compressed_size":1387,"conclusion_name":"Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-c8c19eba9d68881f","role":"sequence","name":"binomial","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-d28ad9841fc73317","claim":"one_signed_coefficients"},{"node_id":"n-d28ad9841fc73317","claim":"hadamard_kernel"},{"node_id":"n-d28ad9841fc73317","claim":"real_rooted"}]}