{"id":"e-1067a49da9ee27d6","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-4b03a2849503b9e9","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":"05e784ec7760cfb6b35971f72179b06ee99fc8f09400b86cb4f20fc97037b026","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":15610,"certificate_compressed_size":3114,"conclusion_name":"shift_x_plus_one of Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 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-f24dc5f77f658331","role":"sequence","name":"Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 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-4b03a2849503b9e9","claim":"one_signed_coefficients"},{"node_id":"n-4b03a2849503b9e9","claim":"hadamard_kernel"},{"node_id":"n-4b03a2849503b9e9","claim":"real_rooted"},{"node_id":"n-4b03a2849503b9e9","claim":"interlacing"}]}