{"id":"e-9f91faf8953a1d7a","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-d0986a7acf60271a","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":"0234f2ba06f53426c69ea06564ba96b9d2049e8973bfce034959815115148637","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":13345,"certificate_compressed_size":2515,"conclusion_name":"shift_x_plus_one of Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-0eba401b117f16a9","role":"sequence","name":"Triangle read by rows: T(n, k) = binomial(n + k, 2*k).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}],"cited_by":[{"node_id":"n-d0986a7acf60271a","claim":"interlacing"},{"node_id":"n-d0986a7acf60271a","claim":"one_signed_coefficients"},{"node_id":"n-d0986a7acf60271a","claim":"hadamard_kernel"},{"node_id":"n-d0986a7acf60271a","claim":"real_rooted"}]}