{"id":"e-e393ec71cd9e88de","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-c73008ca3f0e3c9c","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":"ead236327f462de9fa0f586e5daeb55944a05b608a1deea44a7880c08fd27d91","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":14383,"certificate_compressed_size":2696,"conclusion_name":"shift_x_plus_one of basis_map_A007318 of Triangle T(n, k) of numbers of square lattice walks that start and end at origin after 2*n steps and contain exactly k steps to the east, possibly touching origin at intermediate stages.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-ed80a7592b428c53","role":"sequence","name":"basis_map_A007318 of Triangle T(n, k) of numbers of square lattice walks that start and end at origin after 2*n steps and contain exactly k steps to the east, possibly touching origin at intermediate stages.","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-c73008ca3f0e3c9c","claim":"one_signed_coefficients"},{"node_id":"n-c73008ca3f0e3c9c","claim":"hadamard_kernel"},{"node_id":"n-c73008ca3f0e3c9c","claim":"interlacing"},{"node_id":"n-c73008ca3f0e3c9c","claim":"real_rooted"}]}