{"id":"e-856dc4b7af9f5bee","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-5fc6172728a1754c","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":"ba21337cc277ce0427853760049063d1212abb9ecdf38b9c2d2bfb9d50df56f6","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":14744,"certificate_compressed_size":2814,"conclusion_name":"basis_map_A013609 of Triangular T(n,k) = T(n-1, k) + T(n-1, k-1) + 5*T(n-2, k-1), read by rows.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":3,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-a44eea8c1f4cf4e5","role":"sequence","name":"Triangular T(n,k) = T(n-1, k) + T(n-1, k-1) + 5*T(n-2, k-1), read by rows.","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-5fc6172728a1754c","claim":"real_rooted"},{"node_id":"n-5fc6172728a1754c","claim":"one_signed_coefficients"},{"node_id":"n-5fc6172728a1754c","claim":"hadamard_kernel"},{"node_id":"n-5fc6172728a1754c","claim":"interlacing"}]}