{"id":"e-cb03f5f41d038dd3","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-5d39164c4108e043","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":"617b47798296cd3211a5fdb6bb50be44b8e1bae4d92a348a67a7a0b9348fefb9","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":11671,"certificate_compressed_size":2203,"conclusion_name":"Riordan array (1, 3+x).","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-b0cf60708a253825","role":"sequence","name":"basis-map coefficients of Riordan array (1, 3+x).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"1"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 1]"}],"cited_by":[{"node_id":"n-5d39164c4108e043","claim":"hadamard_kernel"},{"node_id":"n-5d39164c4108e043","claim":"real_rooted"},{"node_id":"n-5d39164c4108e043","claim":"one_signed_coefficients"}]}