{"id":"e-d8c5b2c5645f8542","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-246f6d36a4b25acd","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":"db5b0819be6341ec10e8418b729aceccb18f8a90121dfae1fe4aeb73b7798c33","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":14171,"certificate_compressed_size":2678,"conclusion_name":"basis_map_A013610 of basis_map_A013610 of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k peaks (1 <= k <= n).","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-5b112963159320a9","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k peaks (1 <= k <= n).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}],"cited_by":[{"node_id":"n-246f6d36a4b25acd","claim":"real_rooted"},{"node_id":"n-246f6d36a4b25acd","claim":"one_signed_coefficients"},{"node_id":"n-246f6d36a4b25acd","claim":"hadamard_kernel"},{"node_id":"n-246f6d36a4b25acd","claim":"interlacing"}]}