{"id":"e-7438a94267d81529","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-d9956ec4f2e74f86","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":"6cfe5e17d305fa0d558f5c26c3baa0d1552935d0b140ce7ff2bac7c3cba14ab4","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":14198,"certificate_compressed_size":2658,"conclusion_name":"shift_x_plus_one of Triangle read by rows: T(n,0)=1, T(n,n)=2, T(n,m) = T(n-1,m-1) + Sum_{k=0..m} T(n-1-k,m-k).","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-2496e2f555756362","role":"sequence","name":"Triangle read by rows: T(n,0)=1, T(n,n)=2, T(n,m) = T(n-1,m-1) + Sum_{k=0..m} T(n-1-k,m-k).","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-d9956ec4f2e74f86","claim":"one_signed_coefficients"},{"node_id":"n-d9956ec4f2e74f86","claim":"hadamard_kernel"},{"node_id":"n-d9956ec4f2e74f86","claim":"interlacing"},{"node_id":"n-d9956ec4f2e74f86","claim":"real_rooted"}]}