{"id":"e-1d1ac0038b6609d9","operation":{"index_shift":0,"modulus":2,"operation":"veronese_section","residue":1},"operation_detail":"r=2, j=1, shift 0","conclusion_id":"n-a5420872eb2b873d","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":"c133b089a5b526eed9207db7e0da9eec23439cfc9ccb9b5224020152fba8cddd","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"veronese_section","certificate_raw_size":4531,"certificate_compressed_size":1293,"conclusion_name":"Binomial coefficients C(n,k) with n-k even, read by rows.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-6a690be6f27c7004","role":"sequence","name":"veronese parent of Binomial coefficients C(n,k) with n-k even, 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-a5420872eb2b873d","claim":"real_rooted"},{"node_id":"n-a5420872eb2b873d","claim":"one_signed_coefficients"},{"node_id":"n-a5420872eb2b873d","claim":"hadamard_kernel"}]}