{"id":"e-5892d22641c77abc","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-bc4bccd8525de23a","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":"fcb6e030b25505d6ddc80d4a6e1153423dc3ecfb3634de8bbde280c257bbb2ac","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":16597,"certificate_compressed_size":3217,"conclusion_name":"basis_map_A013609 of derivative of Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first kind in reverse order).","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-89353a17ee1d740a","role":"sequence","name":"derivative of Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first kind in reverse order).","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-bc4bccd8525de23a","claim":"real_rooted"},{"node_id":"n-bc4bccd8525de23a","claim":"one_signed_coefficients"},{"node_id":"n-bc4bccd8525de23a","claim":"hadamard_kernel"},{"node_id":"n-bc4bccd8525de23a","claim":"interlacing"}]}