{"id":"e-a75973ccb9abaa5d","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-75a6475b3e851dc5","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":"8e4384235f3e162723d8d6c80af1c26cb78c501821cb0b931efb9bee6e673d56","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":7643,"certificate_compressed_size":1689,"conclusion_name":"Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-b8bb9d4926920605","role":"sequence","name":"theta_plus_one of 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-75a6475b3e851dc5","claim":"one_signed_coefficients"},{"node_id":"n-75a6475b3e851dc5","claim":"hadamard_kernel"},{"node_id":"n-75a6475b3e851dc5","claim":"real_rooted"}]}