{"id":"e-fe5597de1109704e","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-94470298ec1d4aa5","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":"c52d6a8fd6d6802ec858a446851dd0a13ab028006a9b90edefaadf31c02b83d5","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":14581,"certificate_compressed_size":2818,"conclusion_name":"shift_x_plus_one of Triangle read by rows: T(n,k) is the number of ternary strings of length n+1 with k+1 indispensable digits and a nonzero leading digit, with 0 <= k <= n.","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-539ff84504d2bf3a","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of ternary strings of length n+1 with k+1 indispensable digits and a nonzero leading digit, with 0 <= k <= n.","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-94470298ec1d4aa5","claim":"one_signed_coefficients"},{"node_id":"n-94470298ec1d4aa5","claim":"hadamard_kernel"},{"node_id":"n-94470298ec1d4aa5","claim":"real_rooted"},{"node_id":"n-94470298ec1d4aa5","claim":"interlacing"}]}