{"id":"e-b4f12c588dbcf356","operation":{"index_shift":0,"modulus":2,"operation":"veronese_section","residue":1},"operation_detail":"r=2, j=1, shift 0","conclusion_id":"n-b9bba3062ead0647","premise_count":1,"theorem":"Veronese sections of Polya frequency sequences (Aissen--Schoenberg--Whitney--Edrei): if P is real-rooted with coefficients of one sign, equivalently nonzero with all roots in (-inf, 0], every section V_(r,j)(P) is zero or real-rooted with roots in (-inf, 0]","lean_route":null,"lean_declarations":[],"certificate_hash":"c4b36a9c4ea83f1032de6ed544f6c44cf41bc33c9d4cc4035f3fd4d469c9c4ca","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"veronese_section","certificate_raw_size":9101,"certificate_compressed_size":2441,"conclusion_name":"veronese_section_2_1 of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[{"position":0,"node_id":"n-b9fbfb57d9269e2f","role":"sequence","name":"Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,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-b9bba3062ead0647","claim":"real_rooted"},{"node_id":"n-b9bba3062ead0647","claim":"one_signed_coefficients"},{"node_id":"n-b9bba3062ead0647","claim":"hadamard_kernel"},{"node_id":"n-b9bba3062ead0647","claim":"interlacing"}]}