{"id":"e-ccc293aee69469d4","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-e8f74cc6dbf727f1","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":"f3f3380b905af8a316b8e69f1bfd789e40dcd5aba915030e6fc2fb88d099bf78","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":13219,"certificate_compressed_size":2520,"conclusion_name":"basis_map_A013609 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).","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-a98fae1bf9eeb913","role":"sequence","name":"Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).","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-e8f74cc6dbf727f1","claim":"interlacing"},{"node_id":"n-e8f74cc6dbf727f1","claim":"real_rooted"},{"node_id":"n-e8f74cc6dbf727f1","claim":"one_signed_coefficients"},{"node_id":"n-e8f74cc6dbf727f1","claim":"hadamard_kernel"}]}