{"id":"e-33165e49bd0f30d2","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-f074ae9ffad82cf4","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":"fbeeb3b690d398ba03ccd3821e2ae78b90bd46d92b113b7055c2d524afa81ded","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":14967,"certificate_compressed_size":3006,"conclusion_name":"basis_map_A013609 of Triangle T(n,k) = T(n-1,k) + 3*T(n-2,k-1) for k = 0..floor(n/2), with T(0,0) = 1, T(n,k) = 0 for n or k < 0, read by rows.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-78b97ad8ac29ceb2","role":"sequence","name":"basis_map_A013609 of Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one version of) Fibonacci polynomials.","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-f074ae9ffad82cf4","claim":"real_rooted"},{"node_id":"n-f074ae9ffad82cf4","claim":"one_signed_coefficients"},{"node_id":"n-f074ae9ffad82cf4","claim":"hadamard_kernel"},{"node_id":"n-f074ae9ffad82cf4","claim":"interlacing"}]}