{"id":"e-93941c03d72a750a","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-25c619a616edcf01","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":"da5df896b4fad676fdbdae921320822fae3cff3d3659c052a68011f4dc479098","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":13620,"certificate_compressed_size":2597,"conclusion_name":"basis_map_A007318 of basis_map_A013610 of Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (0<=k<=n).","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-f5286435ed391b4f","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (0<=k<=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-25c619a616edcf01","claim":"real_rooted"},{"node_id":"n-25c619a616edcf01","claim":"one_signed_coefficients"},{"node_id":"n-25c619a616edcf01","claim":"hadamard_kernel"},{"node_id":"n-25c619a616edcf01","claim":"interlacing"}]}