{"id":"e-86bfcfeb53d37b24","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-14506fb9ff389955","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":"e9727d4c60df82d9986973a2698f0dfdc3e74ae4c38952c1feec45237440d112","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":15905,"certificate_compressed_size":3107,"conclusion_name":"shift_x_plus_one of basis_map_A013609 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-83e32d0bc1c10d45","role":"sequence","name":"basis_map_A007318 of Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n containing exactly k peaks but no peaks at level one (n >= 1; 0 <= k <= n-1).","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-14506fb9ff389955","claim":"one_signed_coefficients"},{"node_id":"n-14506fb9ff389955","claim":"hadamard_kernel"},{"node_id":"n-14506fb9ff389955","claim":"interlacing"},{"node_id":"n-14506fb9ff389955","claim":"real_rooted"}]}