{"id":"e-253053961cc5c19e","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-207567e2aa72e118","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":"eff3877b2ecbfba32cc9eafac33dfd01dc6bdb60d124fe0b38cc43face11aad5","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":12392,"certificate_compressed_size":2241,"conclusion_name":"basis_map_A013610 of Triangle read by rows in which T(n,k) is the number of length k chains from (0,0) to (n,n) of the poset [n] X [n] ordered by the product order, 0 <= k <= 2n, n>=0.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-97a17a1dc7998ff8","role":"sequence","name":"Triangle read by rows in which T(n,k) is the number of length k chains from (0,0) to (n,n) of the poset [n] X [n] ordered by the product order, 0 <= k <= 2n, n>=0.","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-207567e2aa72e118","claim":"real_rooted"},{"node_id":"n-207567e2aa72e118","claim":"one_signed_coefficients"},{"node_id":"n-207567e2aa72e118","claim":"hadamard_kernel"}]}