{"id":"e-efe1c301bb1c3fd6","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-ca6d44114ccba1b3","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":"4f2eabe25e0882fa73b3ad8387074ef887f27a7fb43b274e7dde2e67dde00097","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":6865,"certificate_compressed_size":1393,"conclusion_name":"basis_map_A013609 of basis_map_A013610 of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-cb064c72efa341c9","role":"sequence","name":"basis_map_A013609 of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","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-ca6d44114ccba1b3","claim":"real_rooted"},{"node_id":"n-ca6d44114ccba1b3","claim":"one_signed_coefficients"},{"node_id":"n-ca6d44114ccba1b3","claim":"hadamard_kernel"}]}