{"id":"e-b19ded6a55d5da85","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-1852b727a5d6b55d","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":"fe9026e54e174ca3eb1ff199a8dbbece8b18bd044b4a69ea31fb0e2246e41f95","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":13345,"certificate_compressed_size":2520,"conclusion_name":"basis_map_A013609 of Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-0eba401b117f16a9","role":"sequence","name":"Triangle read by rows: T(n, k) = binomial(n + k, 2*k).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}],"cited_by":[{"node_id":"n-1852b727a5d6b55d","claim":"interlacing"},{"node_id":"n-1852b727a5d6b55d","claim":"real_rooted"},{"node_id":"n-1852b727a5d6b55d","claim":"one_signed_coefficients"},{"node_id":"n-1852b727a5d6b55d","claim":"hadamard_kernel"}]}