{"id":"e-bc7c2cf81eba631a","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-e5e648370a9e4459","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":"67eb6bee3ae2027da3dee1b6c84d57427fe938edbd12f6674dc9610dc03b1aab","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":10663,"certificate_compressed_size":2344,"conclusion_name":"Triangular array: the fusion of polynomial sequences P and Q given by p(n,x) = (x+2)^n and q(n,x) = (2*x+1)^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-685ab5842fbad7f4","role":"sequence","name":"Triangular matrix arising in enumeration of catafusenes, read by rows.","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-e5e648370a9e4459","claim":"one_signed_coefficients"},{"node_id":"n-e5e648370a9e4459","claim":"hadamard_kernel"},{"node_id":"n-e5e648370a9e4459","claim":"real_rooted"},{"node_id":"n-e5e648370a9e4459","claim":"interlacing"}]}