{"id":"e-8cb7f9f8489fa121","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-e5bdf85b655e989a","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":"c199ab05fe704e2e5d0319fbe6eabac93d73d1dd7bd34746c5d1995d82ea8ab0","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":14522,"certificate_compressed_size":2901,"conclusion_name":"shift_x_plus_one of Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4^k * C(n+1,2*k+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-cf38558d99f4fe37","role":"sequence","name":"Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4^k * C(n+1,2*k+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-e5bdf85b655e989a","claim":"one_signed_coefficients"},{"node_id":"n-e5bdf85b655e989a","claim":"hadamard_kernel"},{"node_id":"n-e5bdf85b655e989a","claim":"real_rooted"},{"node_id":"n-e5bdf85b655e989a","claim":"interlacing"}]}