{"id":"e-7179f13fe40c0d1d","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-d8a20bc01f7e3b27","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":"2afb9fd44b353df484dba59db5699340258d3c1ac11e1f45cc696ac3919b0faa","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":14534,"certificate_compressed_size":2905,"conclusion_name":"shift_x_plus_one of gamma_polynomial of Triangle read by rows: T(n, k) = binomial(2n, 2k).","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-4399296e302f2e9b","role":"sequence","name":"gamma_polynomial of Triangle read by rows: T(n, k) = binomial(2n, 2k).","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-d8a20bc01f7e3b27","claim":"one_signed_coefficients"},{"node_id":"n-d8a20bc01f7e3b27","claim":"hadamard_kernel"},{"node_id":"n-d8a20bc01f7e3b27","claim":"real_rooted"},{"node_id":"n-d8a20bc01f7e3b27","claim":"interlacing"}]}