{"id":"e-23eee42afd6ddf98","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-e7973e04614ef858","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":"a1333b1604d11592b04b5455ffd6896cd4c05f813f2d0f8efaddcd2601c21c1c","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":6857,"certificate_compressed_size":1393,"conclusion_name":"shift_x_plus_one of basis_map_A013609 of Triangle of coefficients in expansion of (1+3*x)^n.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-67dfd62e07739d13","role":"sequence","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j).","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-e7973e04614ef858","claim":"one_signed_coefficients"},{"node_id":"n-e7973e04614ef858","claim":"hadamard_kernel"},{"node_id":"n-e7973e04614ef858","claim":"real_rooted"}]}