{"id":"e-e2abcf6e1ca1607c","operation":{"index_shift":0,"modulus":2,"operation":"veronese_section","residue":1},"operation_detail":"r=2, j=1, shift 0","conclusion_id":"n-0ed898288ff872d1","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":"3ec290fb9e5909bd1114b3f49707852e43e011a4c9e52c0b009cef2a1545e329","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"veronese_section","certificate_raw_size":7750,"certificate_compressed_size":2208,"conclusion_name":"veronese_section_2_1 of Exponential Riordan array [x*exp(x),x].","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-7b5887f058255aa1","role":"sequence","name":"Triangle of denominators in Leibniz's Harmonic Triangle a(n,k), n >= 1, 1 <= k <= n.","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-0ed898288ff872d1","claim":"real_rooted"},{"node_id":"n-0ed898288ff872d1","claim":"one_signed_coefficients"},{"node_id":"n-0ed898288ff872d1","claim":"hadamard_kernel"}]}