{"id":"e-cf8568343e28676f","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-f1591709bf5cf41b","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":"1ba7f8c242e7c6e74da26714cd6d4aa26428f1870ad93d18a57d1c5d6287b7bb","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":11767,"certificate_compressed_size":2134,"conclusion_name":"derivative of Even-numbered rows of Pascal's triangle.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-e75bc09e82ca4254","role":"sequence","name":"derivative 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-f1591709bf5cf41b","claim":"real_rooted"},{"node_id":"n-f1591709bf5cf41b","claim":"one_signed_coefficients"},{"node_id":"n-f1591709bf5cf41b","claim":"hadamard_kernel"}]}