{"id":"e-1e55567759b4d30f","operation":{"index_shift":0,"modulus":3,"operation":"veronese_section","residue":1},"operation_detail":"r=3, j=1, shift 0","conclusion_id":"n-e94410689654e39c","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":"88ef48c3882eeebc4925b3e86790bfdd862cede81b0a06a7feb81606b02900a6","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"veronese_section","certificate_raw_size":4969,"certificate_compressed_size":1308,"conclusion_name":"Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-07e2f0a7469aac14","role":"sequence","name":"veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}],"cited_by":[{"node_id":"n-e94410689654e39c","claim":"real_rooted"},{"node_id":"n-e94410689654e39c","claim":"one_signed_coefficients"},{"node_id":"n-e94410689654e39c","claim":"hadamard_kernel"}]}