{"id":"e-a9ccf5f34daf84e6","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-40465177841ad1ac","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":"44287bcc9562f8ce4b4ffd37b6c48d442dd33e6bd2317a77e4c3af40d1942684","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":14967,"certificate_compressed_size":2906,"conclusion_name":"shift_x_plus_one of basis_map_A013609 of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","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-c63944481deaae45","role":"sequence","name":"basis_map_A007318 of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.","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-40465177841ad1ac","claim":"one_signed_coefficients"},{"node_id":"n-40465177841ad1ac","claim":"hadamard_kernel"},{"node_id":"n-40465177841ad1ac","claim":"real_rooted"},{"node_id":"n-40465177841ad1ac","claim":"interlacing"}]}