{"id":"e-83332bc01455e815","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-cb5d49e71e46f2ed","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":"f6eb1a0b4a0beb21e5dd6353d21cfc51d40693adcee19838a57a6fc481882730","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":7409,"certificate_compressed_size":1613,"conclusion_name":"Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-fbb4b9430d9de872","role":"sequence","name":"Triangle read by rows: row n consists of n zeros followed by 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-cb5d49e71e46f2ed","claim":"one_signed_coefficients"},{"node_id":"n-cb5d49e71e46f2ed","claim":"hadamard_kernel"},{"node_id":"n-cb5d49e71e46f2ed","claim":"real_rooted"}]}