{"id":"e-8562aab4e97e2e3f","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-8229dbdd6a4b5cd9","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":"1da6f7ac835e04c2426e066aa35ec06ab0d84df6ca9a9549a9f473f24157ec78","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":7414,"certificate_compressed_size":1617,"conclusion_name":"Triangle read by rows, s_3(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-260e022530e5976c","role":"sequence","name":"Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 3*x)^(-1/3), (-1/3)*log(1 - 3*x)). A generalized Stirling1 triangle.","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-8229dbdd6a4b5cd9","claim":"one_signed_coefficients"},{"node_id":"n-8229dbdd6a4b5cd9","claim":"hadamard_kernel"},{"node_id":"n-8229dbdd6a4b5cd9","claim":"real_rooted"}]}