{"id":"e-eae4e58fb0906170","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-aeb67fdaad2bc42e","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":"d8088065f03d026a522e8396e7b86c60366afeab0fbaf9eb575aa68bdf2b7c24","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":10652,"certificate_compressed_size":2336,"conclusion_name":"shift_x_plus_one of Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 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-569a801b809b6abd","role":"sequence","name":"Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 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-aeb67fdaad2bc42e","claim":"interlacing"},{"node_id":"n-aeb67fdaad2bc42e","claim":"one_signed_coefficients"},{"node_id":"n-aeb67fdaad2bc42e","claim":"hadamard_kernel"},{"node_id":"n-aeb67fdaad2bc42e","claim":"real_rooted"}]}