{"id":"e-32ae8a16e295b946","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-9ac71617f56e595e","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":"13a28446a7f42bdaefef2acca9341c11b060e06c3907cfb9ac0c4b7d02f4dd23","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":13465,"certificate_compressed_size":2549,"conclusion_name":"basis_map_A013609 of Matrix product of unsigned Lah-triangle |A008297(n,k)| and unsigned Stirling1-triangle |A008275(n,k)|.","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-6ace63f6e12c5046","role":"sequence","name":"Matrix product of unsigned Lah-triangle |A008297(n,k)| and unsigned Stirling1-triangle |A008275(n,k)|.","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-9ac71617f56e595e","claim":"interlacing"},{"node_id":"n-9ac71617f56e595e","claim":"real_rooted"},{"node_id":"n-9ac71617f56e595e","claim":"one_signed_coefficients"},{"node_id":"n-9ac71617f56e595e","claim":"hadamard_kernel"}]}