{"id":"e-0f82e3c94102b0bd","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-ea5fd95b906f5bdb","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":"c97559122decc17b1e1b65226ec94658c421009c0653d6e09b96ba118340575b","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":14066,"certificate_compressed_size":2700,"conclusion_name":"basis_map_A013609 of Triangle read by rows: T(n,k) is the number of k-matchings of the corona K'(n) of the complete graph K(n) and the complete graph K(1); in other words, K'(n) is the graph constructed from K(n) by adding for each vertex v a new vertex v' and the edge vv'.","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-07e5356ad37f61ea","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of k-matchings of the corona K'(n) of the complete graph K(n) and the complete graph K(1); in other words, K'(n) is the graph constructed from K(n) by adding for each vertex v a new vertex v' and the edge vv'.","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-ea5fd95b906f5bdb","claim":"real_rooted"},{"node_id":"n-ea5fd95b906f5bdb","claim":"one_signed_coefficients"},{"node_id":"n-ea5fd95b906f5bdb","claim":"hadamard_kernel"},{"node_id":"n-ea5fd95b906f5bdb","claim":"interlacing"}]}