{"id":"e-404109e50a21afd4","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-6a5af5e1168bb679","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":"edc5082fe1f50541bff18f0c35b8edc80fcb765f441d78e4a3322fea811a5812","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":6855,"certificate_compressed_size":1384,"conclusion_name":"Triangle of coefficients in expansion of (1+3*x)^n.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-6fea5aff4f44fa54","role":"sequence","name":"a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.","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-6a5af5e1168bb679","claim":"one_signed_coefficients"},{"node_id":"n-6a5af5e1168bb679","claim":"hadamard_kernel"},{"node_id":"n-6a5af5e1168bb679","claim":"real_rooted"}]}