{"id":"e-e6ead7ccbecb2d32","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-054881b8a1092a7d","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":"a65b2f6e8572d585cb5171b1cef845ea3fccc433e625302ac79a42dd265bf0d1","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":7414,"certificate_compressed_size":1617,"conclusion_name":"basis_map_A013611 of Triangle T(n, k) read by rows: T(n, k) = Sum_{j=0..n} binomial(j, n-k) * |Stirling1(n, n-j)|.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-849de7332df28657","role":"sequence","name":"Triangle T(n, k) read by rows: T(n, k) = Sum_{j=0..n} binomial(j, n-k) * |Stirling1(n, n-j)|.","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-054881b8a1092a7d","claim":"real_rooted"},{"node_id":"n-054881b8a1092a7d","claim":"one_signed_coefficients"},{"node_id":"n-054881b8a1092a7d","claim":"hadamard_kernel"}]}