{"id":"e-125a67b606fd2347","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-d5dd236ae5fa65ba","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":"09327501c6d30bcf8d2064907257dec3cda2c43593f48e82db973ffcc0ab50d0","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":13449,"certificate_compressed_size":2553,"conclusion_name":"basis_map_A013609 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-8eb9bb3a09f0b7fd","role":"sequence","name":"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-d5dd236ae5fa65ba","claim":"real_rooted"},{"node_id":"n-d5dd236ae5fa65ba","claim":"one_signed_coefficients"},{"node_id":"n-d5dd236ae5fa65ba","claim":"hadamard_kernel"},{"node_id":"n-d5dd236ae5fa65ba","claim":"interlacing"}]}