{"id":"e-666ec6ed045ae37a","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-23c9ebec99e56671","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":"362088e80e800f3b77d3a80c0f9a8d73921dbac1623c60493f20c231b34e80d0","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":6877,"certificate_compressed_size":1395,"conclusion_name":"shift_x_plus_one of basis_map_A013609 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-8c0973a753fcbe28","role":"sequence","name":"basis_map_A007318 of Even-numbered rows of Pascal's triangle.","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-23c9ebec99e56671","claim":"one_signed_coefficients"},{"node_id":"n-23c9ebec99e56671","claim":"hadamard_kernel"},{"node_id":"n-23c9ebec99e56671","claim":"real_rooted"}]}