{"id":"e-6521fdb104380f39","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-1d7b3a8d2533ff46","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":"8109c7ecec220a856a12d6d6753f8d2018fba3fab4cf8fd1f96197fb239a3148","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":15239,"certificate_compressed_size":2814,"conclusion_name":"derivative of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.","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-af521840f49b6d03","role":"sequence","name":"derivative of Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A level is a pair of identical consecutive letters.","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-1d7b3a8d2533ff46","claim":"real_rooted"},{"node_id":"n-1d7b3a8d2533ff46","claim":"one_signed_coefficients"},{"node_id":"n-1d7b3a8d2533ff46","claim":"hadamard_kernel"},{"node_id":"n-1d7b3a8d2533ff46","claim":"interlacing"}]}