{"id":"e-ca0c9c4bd1e03317","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-f5dd8e1ee54613a3","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":"8fff4d82d6a3daf39a8dffeba5203d7a0280794d6dddc7f03f5a266cdd9c2250","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":14502,"certificate_compressed_size":2878,"conclusion_name":"basis_map_A013611 of Triangle T(n,k) read by rows: associated Stirling numbers of first kind (n >= 2, 1 <= k <= floor(n/2)).","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-f13c6b611f4ce3bb","role":"sequence","name":"Triangle T(n,k) read by rows: associated Stirling numbers of first kind (n >= 2, 1 <= k <= floor(n/2)).","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-f5dd8e1ee54613a3","claim":"real_rooted"},{"node_id":"n-f5dd8e1ee54613a3","claim":"one_signed_coefficients"},{"node_id":"n-f5dd8e1ee54613a3","claim":"hadamard_kernel"},{"node_id":"n-f5dd8e1ee54613a3","claim":"interlacing"}]}