{"id":"e-d0c14addeb793323","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-7d332fcb7bf9a0f7","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":"26ac1ed4706fcacf6748d0e4d0fde57744d00e74cf4c621345276aba647daf06","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":10694,"certificate_compressed_size":2400,"conclusion_name":"Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pieces of 1 X 2 tiles (0 <= k <= floor(n/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-75ccd0453a9b6cec","role":"sequence","name":"Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one version of) Fibonacci polynomials.","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-7d332fcb7bf9a0f7","claim":"one_signed_coefficients"},{"node_id":"n-7d332fcb7bf9a0f7","claim":"hadamard_kernel"},{"node_id":"n-7d332fcb7bf9a0f7","claim":"real_rooted"},{"node_id":"n-7d332fcb7bf9a0f7","claim":"interlacing"}]}