{"id":"e-4b5346f304dc9a08","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-34a4e524e6a5040f","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":"2bc96f9086ceb0a37223365d424acf919463713135241dcba0320b82cd0be1d1","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":10358,"certificate_compressed_size":2304,"conclusion_name":"shift_x_plus_one of basis_map_A013609 of Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/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-623036dc25b33b8c","role":"sequence","name":"basis_map_A013609 of Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).","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-34a4e524e6a5040f","claim":"one_signed_coefficients"},{"node_id":"n-34a4e524e6a5040f","claim":"hadamard_kernel"},{"node_id":"n-34a4e524e6a5040f","claim":"interlacing"},{"node_id":"n-34a4e524e6a5040f","claim":"real_rooted"}]}