{"id":"e-cd3fa89de2aead53","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-51a2369b25b8e3ba","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":"76cb2efb0bc62db89c8196124f74a985c332791a1750029b0ec13a502177ce37","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":11582,"certificate_compressed_size":2147,"conclusion_name":"shift_x_plus_one of Triangle read by rows: T(n,k) = binomial(k,n-k).","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-21854f4257effb01","role":"sequence","name":"Triangle read by rows: T(n,k) = binomial(k,n-k).","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-51a2369b25b8e3ba","claim":"one_signed_coefficients"},{"node_id":"n-51a2369b25b8e3ba","claim":"hadamard_kernel"},{"node_id":"n-51a2369b25b8e3ba","claim":"real_rooted"}]}