{"id":"e-fc45cb3590dab8d3","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-c5818ec04781ce60","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":"b84ac713a49eae3f2246eed5ade4842d1296bb08f976c5bd701a3276b4cf21e7","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":7455,"certificate_compressed_size":1622,"conclusion_name":"basis_map_A007318 of Triangle read by rows, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + (n-1)^2*T(n-1, k), for 0 <= k <= n.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-54284d24a094e2d3","role":"sequence","name":"Triangle read by rows, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + (n-1)^2*T(n-1, k), for 0 <= k <= n.","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-c5818ec04781ce60","claim":"real_rooted"},{"node_id":"n-c5818ec04781ce60","claim":"one_signed_coefficients"},{"node_id":"n-c5818ec04781ce60","claim":"hadamard_kernel"}]}