{"id":"e-01cd83cc52a7b84d","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-390f0d4f48e5b502","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":"8d8675641d0b475aa78f85fca3d3a787b8df4e301e7b48f44fcd32a4e9dd1502","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":11876,"certificate_compressed_size":2217,"conclusion_name":"basis_map_A013611 of Triangle read by rows: T(n,k) is the number of flattened Catalan words of length n with exactly k runs of descents.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-5576233c61c40153","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of flattened Catalan words of length n with exactly k runs of descents.","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-390f0d4f48e5b502","claim":"real_rooted"},{"node_id":"n-390f0d4f48e5b502","claim":"one_signed_coefficients"},{"node_id":"n-390f0d4f48e5b502","claim":"hadamard_kernel"}]}