{"id":"e-169cfb9127a1a777","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-3897d3bccfbe11da","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":"9618c2d63a8c393a533f07ecc3130e86fc94c7c0c102c53b613a7acd7408c881","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":14558,"certificate_compressed_size":2761,"conclusion_name":"basis_map_A013609 of Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1).","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-b1b19325e0094118","role":"sequence","name":"Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 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-3897d3bccfbe11da","claim":"real_rooted"},{"node_id":"n-3897d3bccfbe11da","claim":"one_signed_coefficients"},{"node_id":"n-3897d3bccfbe11da","claim":"hadamard_kernel"},{"node_id":"n-3897d3bccfbe11da","claim":"interlacing"}]}