{"id":"e-16a835f21d117a04","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-d9640fb05f96c50b","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":"06112203c6494ab76efac8d454e12766959aab2ff2bc2c84161e33461c7910f8","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":11876,"certificate_compressed_size":2111,"conclusion_name":"shift_x_plus_one of Triangle read by rows: T(n, k) (0<=k<=2n) is the number of Delannoy paths of length n, having k steps.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-e4d248920a317259","role":"sequence","name":"Triangle read by rows: T(n, k) (0<=k<=2n) is the number of Delannoy paths of length n, having k steps.","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-d9640fb05f96c50b","claim":"one_signed_coefficients"},{"node_id":"n-d9640fb05f96c50b","claim":"hadamard_kernel"},{"node_id":"n-d9640fb05f96c50b","claim":"real_rooted"}]}