{"id":"e-719615516772b132","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-c27b11147fce8bfa","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":"0c60623b0d674a8c81ca0bd6dddf8ce7cca19e9af9509ad4715d42b020833e52","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":11588,"certificate_compressed_size":2153,"conclusion_name":"Riordan array (1+2x,x(1+x)).","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-d2980b44b9e1faf3","role":"sequence","name":"basis-map coefficients of Riordan array (1+2x,x(1+x)).","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"1"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 1]"}],"cited_by":[{"node_id":"n-c27b11147fce8bfa","claim":"one_signed_coefficients"},{"node_id":"n-c27b11147fce8bfa","claim":"hadamard_kernel"},{"node_id":"n-c27b11147fce8bfa","claim":"real_rooted"}]}