{"id":"e-9e3965a70175b87c","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-eb97e93fd65cf0d3","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"bca915750e0d9c3160474da14f8adfbd92118ef4667b7675f222cf0e651df2fd","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":14143,"certificate_compressed_size":2769,"conclusion_name":"shift_x_plus_one of T(n, k) = binomial(n, k)*hypergeom([(1 - n)/2, -n/2], [1], 4).","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-63b9a6ee72661305","role":"sequence","name":"T(n, k) = binomial(n, k)*hypergeom([(1 - n)/2, -n/2], [1], 4).","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-eb97e93fd65cf0d3","claim":"one_signed_coefficients"},{"node_id":"n-eb97e93fd65cf0d3","claim":"hadamard_kernel"},{"node_id":"n-eb97e93fd65cf0d3","claim":"real_rooted"}]}