{"id":"e-7d33761108e62398","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-2494c903d5dafb67","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":"615a2676e6eed9aa6def5229f813c6aacfa94598a99b1e3f83ddb32aa218be86","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":15237,"certificate_compressed_size":2810,"conclusion_name":"derivative of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-fc09c8f5d575becf","role":"sequence","name":"derivative of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","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-2494c903d5dafb67","claim":"real_rooted"},{"node_id":"n-2494c903d5dafb67","claim":"one_signed_coefficients"},{"node_id":"n-2494c903d5dafb67","claim":"hadamard_kernel"},{"node_id":"n-2494c903d5dafb67","claim":"interlacing"}]}