{"id":"e-8d8fb2debc89383b","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-6154960a0891a0b7","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":"5d0b78ee4830b54f35f1110a5e1757a1f49f0c378131fe6d898c60ce1ba9c8e9","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":14011,"certificate_compressed_size":2694,"conclusion_name":"basis_map_A007318 of Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1).","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-14dfd91463cae288","role":"sequence","name":"Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-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-6154960a0891a0b7","claim":"one_signed_coefficients"},{"node_id":"n-6154960a0891a0b7","claim":"hadamard_kernel"},{"node_id":"n-6154960a0891a0b7","claim":"interlacing"},{"node_id":"n-6154960a0891a0b7","claim":"real_rooted"}]}