{"id":"e-d0341b4ac1663435","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-3ca9019ed589d9c3","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":"94332105b9f51a7ea74bbc60b8f6554fa033c509fbe1b2621b56dbb2d13e08d9","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":14120,"certificate_compressed_size":2702,"conclusion_name":"basis_map_A007318 of Triangle T(n, k) = coefficients of p(x, n) where p(x,n) = (x+1)*p(x, n-1) + n*(n-1)*x*p(x, n-2), read by rows.","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-08ac2c6d9914754c","role":"sequence","name":"Triangle T(n, k) = coefficients of p(x, n) where p(x,n) = (x+1)*p(x, n-1) + n*(n-1)*x*p(x, n-2), read by rows.","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-3ca9019ed589d9c3","claim":"real_rooted"},{"node_id":"n-3ca9019ed589d9c3","claim":"one_signed_coefficients"},{"node_id":"n-3ca9019ed589d9c3","claim":"hadamard_kernel"},{"node_id":"n-3ca9019ed589d9c3","claim":"interlacing"}]}