{"id":"e-37edf5cceba94f47","operation":{"map":"A038327","operation":"basis_map"},"operation_detail":"A038327","conclusion_id":"n-c72653d9ab0ee8be","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":"ed2ed6a297e8a94af1b8c92494baa9f8bec2cd17d1f5bc0137f5e864755d9ce1","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":12089,"certificate_compressed_size":2309,"conclusion_name":"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.","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-383c73dff2796e24","role":"sequence","name":"basis-map coefficients 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":"12"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 12]"}],"cited_by":[{"node_id":"n-c72653d9ab0ee8be","claim":"one_signed_coefficients"},{"node_id":"n-c72653d9ab0ee8be","claim":"hadamard_kernel"},{"node_id":"n-c72653d9ab0ee8be","claim":"real_rooted"}]}