{"id":"e-484d449f4d0f99c5","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-fe7b03c8bdb96043","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":"b0cef0bec03ed03fa8aee02fd86463b2e327e96d30d577a4c1f68315eb00066a","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":13197,"certificate_compressed_size":2501,"conclusion_name":"basis_map_A007318 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","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-79986ab5023c2764","role":"sequence","name":"Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","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-fe7b03c8bdb96043","claim":"real_rooted"},{"node_id":"n-fe7b03c8bdb96043","claim":"one_signed_coefficients"},{"node_id":"n-fe7b03c8bdb96043","claim":"hadamard_kernel"},{"node_id":"n-fe7b03c8bdb96043","claim":"interlacing"}]}