{"id":"e-c4975218c7e03a01","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-25833e4dbf2c97ec","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":"d9f2caf8d6bb40cff8523b0d3a823671ca6a3c62acf74638eea88f73ecaa8c44","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":16334,"certificate_compressed_size":3212,"conclusion_name":"basis_map_A013611 of Triangle read by rows with T(n,0) = 1 for n >= 0 and T(n,k) = binomial(n-1,k-1)*(2*k*(n-k) + n)/k for 0 < k <= n.","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-55c53861db6c014a","role":"sequence","name":"Triangle read by rows with T(n,0) = 1 for n >= 0 and T(n,k) = binomial(n-1,k-1)*(2*k*(n-k) + n)/k 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-25833e4dbf2c97ec","claim":"real_rooted"},{"node_id":"n-25833e4dbf2c97ec","claim":"one_signed_coefficients"},{"node_id":"n-25833e4dbf2c97ec","claim":"hadamard_kernel"},{"node_id":"n-25833e4dbf2c97ec","claim":"interlacing"}]}