{"id":"e-056af2fe159cca9f","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-86243365a11d35b7","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":"366937057395a89492a27f387f22e53bc3cf334af14c4192252c68db27858b8a","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":13770,"certificate_compressed_size":2613,"conclusion_name":"basis_map_A007318 of basis_map_A007318 of Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/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-cca535f6115f5fc0","role":"sequence","name":"basis_map_A007318 of Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/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-86243365a11d35b7","claim":"real_rooted"},{"node_id":"n-86243365a11d35b7","claim":"one_signed_coefficients"},{"node_id":"n-86243365a11d35b7","claim":"hadamard_kernel"},{"node_id":"n-86243365a11d35b7","claim":"interlacing"}]}