{"id":"e-30ed1b2c5535cdc0","operation":{"index_shift":0,"modulus":2,"operation":"veronese_section","residue":0},"operation_detail":"r=2, j=0, shift 0","conclusion_id":"n-1427d0a7d0609176","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":"4c5a070e62e5627d48f0a074752be5ebd962e42309f7d51054cbcb0abe8d03b1","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"veronese_section","certificate_raw_size":7762,"certificate_compressed_size":2205,"conclusion_name":"veronese_section_2_0 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-1427d0a7d0609176","claim":"real_rooted"},{"node_id":"n-1427d0a7d0609176","claim":"one_signed_coefficients"},{"node_id":"n-1427d0a7d0609176","claim":"hadamard_kernel"}]}