{"id":"e-3fa6b391d3cb5b3f","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-cf5d6ef2e8520cdb","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":"a20669244371d1a56fb05d300e22da29bb11dc380d689d85bb0f512394ebad9d","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":10244,"certificate_compressed_size":2289,"conclusion_name":"shift_x_plus_one of basis_map_A013611 of Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A generalized Stirling1 triangle.","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-bf0f86dc9ba61887","role":"sequence","name":"basis_map_A013609 of Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 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-cf5d6ef2e8520cdb","claim":"one_signed_coefficients"},{"node_id":"n-cf5d6ef2e8520cdb","claim":"hadamard_kernel"},{"node_id":"n-cf5d6ef2e8520cdb","claim":"interlacing"},{"node_id":"n-cf5d6ef2e8520cdb","claim":"real_rooted"}]}