{"id":"e-f6175c0eb6a5fe0b","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-a88d7af0cffbf501","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":"06b97dfe91ef96483f224eb5a17c90ca65a24ec53adc614adcc6a43bfc0a7e0c","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":6857,"certificate_compressed_size":1393,"conclusion_name":"reversal of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-53ad300b8c449980","role":"sequence","name":"basis_map_A013610 of Triangle of coefficients in expansion of (1+2*x)^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-a88d7af0cffbf501","claim":"real_rooted"},{"node_id":"n-a88d7af0cffbf501","claim":"one_signed_coefficients"},{"node_id":"n-a88d7af0cffbf501","claim":"hadamard_kernel"}]}