{"id":"e-7b3d5e46dee7f2c5","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-8d96419786c3ee2a","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":"e0c6a576d399d27c71ebfc64ac4c66413de3520b96cbd5dfb4c7963fc0b695a9","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":12290,"certificate_compressed_size":2331,"conclusion_name":"Triangle read by rows: T(n,k) = (2^(n+1)-1)*binomial(n,k).","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-6394a556f7ad3cd4","role":"sequence","name":"Triangle read by rows, (n-1) zeros followed by 2^n - 1.","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-8d96419786c3ee2a","claim":"one_signed_coefficients"},{"node_id":"n-8d96419786c3ee2a","claim":"hadamard_kernel"},{"node_id":"n-8d96419786c3ee2a","claim":"real_rooted"}]}