{"id":"e-ab1c685224913bdf","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-647a94c6d9dbf531","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":"3b803ac54c4511a2f8341fe3ad0a0a83265fdf89e6a08c788a78b084e6caf9aa","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":14395,"certificate_compressed_size":2714,"conclusion_name":"basis_map_A013610 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-647a94c6d9dbf531","claim":"real_rooted"},{"node_id":"n-647a94c6d9dbf531","claim":"one_signed_coefficients"},{"node_id":"n-647a94c6d9dbf531","claim":"hadamard_kernel"},{"node_id":"n-647a94c6d9dbf531","claim":"interlacing"}]}