{"id":"e-70173e467e16020f","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-e3d901814302c059","premise_count":0,"theorem":"constant current multiplier: P_0 = 1, P_1 = s, P_(k+2) = a P_(k+1) + (b t + c) P_k with a, s >= 0, b > 0.  P_k(t) = Q_k(b t + c) for the rows Q_k with b = 1, c = 0, whose map sends PF inputs to PF outputs (an interlacing-preserving 2 x 2 matrix, Branden, arXiv:1410.6601, Theorem 8.5), so every PF input has a real-rooted image with roots <= -c/b; PF inputs (also those with roots in any [-R, 0]) have PF images if and only if every P_k(0) >= 0, if and only if a^2 + 4c >= 0 and (2s >= a or c >= s^2 - a s) (RealRooted orderTwoBasis_pf_equivalences, basisTransform_orderTwoBasis_preserves_bounded_pf_iff)","lean_route":null,"lean_declarations":[],"certificate_hash":"1462e95f624943e4070baf80d1ffaecf9194732c2b95ef339772fd391f36a05a","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map_theorem","certificate_raw_size":3178,"certificate_compressed_size":1059,"conclusion_name":"Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X 2 tiles and 3n-4k pieces of 1 X 1 tiles (0 <= k <= floor(n/2)).","claims":[{"claim":"basis_map_preserver","input_interval":"(-inf, 0]","output_interval":"(-inf, 0]","statement":"every PF input has a real-rooted image with every root <= -c/b = 0"},{"claim":"basis_map_preserver","input_interval":"(-inf, 0]","output_interval":"(-inf, 0]","statement":"every PF input has a PF image (zero allowed)"}],"premises":[],"cited_by":[{"node_id":"n-e3d901814302c059","claim":"basis_map_preserver"}]}