{"id":"e-5dce914ed1d38b33","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-23c8d0096dc333e5","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":"b760365b4257394a53f2d35a8143703203f9f3143926056f2da787ac8d9dc751","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":11326,"certificate_compressed_size":2022,"conclusion_name":"shift_x_plus_one of theta_plus_one of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-f3766a15d78768ea","role":"sequence","name":"theta_plus_one of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0","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-23c8d0096dc333e5","claim":"one_signed_coefficients"},{"node_id":"n-23c8d0096dc333e5","claim":"hadamard_kernel"},{"node_id":"n-23c8d0096dc333e5","claim":"real_rooted"}]}