{"id":"e-1e485ec5a5333828","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-7d65d13b7440528c","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":"ee7e68394838af0340b554b14ce65939718c94ac659e8d8bfe28ca5f1365d7af","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":10813,"certificate_compressed_size":2376,"conclusion_name":"basis_map_A013609 of Triangle T(n,k) = C(n+k-1,k)*C(2*n-1,n-k).","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-9a3e4fcd8c6a0286","role":"sequence","name":"Triangle T(n,k) = C(n+k-1,k)*C(2*n-1,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-7d65d13b7440528c","claim":"real_rooted"},{"node_id":"n-7d65d13b7440528c","claim":"one_signed_coefficients"},{"node_id":"n-7d65d13b7440528c","claim":"hadamard_kernel"},{"node_id":"n-7d65d13b7440528c","claim":"interlacing"}]}