{"id":"e-ef158e576c5477a7","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-169aeabe58b22027","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":"d2c7de82fe7840e16e5fd545aa07395749b8c66e352c9dfaa7d51b8a263768f2","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":13984,"certificate_compressed_size":2618,"conclusion_name":"basis_map_A013610 of Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan triangle.","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-c2eaf5212a4dec13","role":"sequence","name":"Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan triangle.","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-169aeabe58b22027","claim":"interlacing"},{"node_id":"n-169aeabe58b22027","claim":"real_rooted"},{"node_id":"n-169aeabe58b22027","claim":"one_signed_coefficients"},{"node_id":"n-169aeabe58b22027","claim":"hadamard_kernel"}]}