{"id":"e-cb0f44f8cbe9fc7d","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-eba41671c715d7a2","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":"f0888b58a8ce16d75b3bae5d91916d6a1a1e532654336d84c243cbfda4cedc57","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":13636,"certificate_compressed_size":2583,"conclusion_name":"basis_map_A013611 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-c85abc56078671cc","role":"sequence","name":"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-eba41671c715d7a2","claim":"real_rooted"},{"node_id":"n-eba41671c715d7a2","claim":"one_signed_coefficients"},{"node_id":"n-eba41671c715d7a2","claim":"hadamard_kernel"},{"node_id":"n-eba41671c715d7a2","claim":"interlacing"}]}