{"id":"e-7573ceab41740b45","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-7ba9c2ffd0ca797f","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":"10e94101a08bde77248bfe3fb19d093aa7c8ad0872412b88ed0b04a90a9b0d68","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":11868,"certificate_compressed_size":2204,"conclusion_name":"shift_x_plus_one of Triangle read by rows: T(n,k) is the number of binary words of length n with k strictly increasing runs, for 0<=k<=n.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-e8e64448d4c6498a","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of binary words of length n with k strictly increasing runs, for 0<=k<=n.","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-7ba9c2ffd0ca797f","claim":"one_signed_coefficients"},{"node_id":"n-7ba9c2ffd0ca797f","claim":"hadamard_kernel"},{"node_id":"n-7ba9c2ffd0ca797f","claim":"real_rooted"}]}