{"id":"e-420bffb5d93b204a","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-0cf41fd0b9103ec6","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":"b5fa7e3e0bfe261b4ee406851d715360d11cbc4ef4b5a61de7a80b2507977079","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":15424,"certificate_compressed_size":2864,"conclusion_name":"shift_x_plus_one of derivative of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","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-f6b66860ba98a2c1","role":"sequence","name":"derivative 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-0cf41fd0b9103ec6","claim":"one_signed_coefficients"},{"node_id":"n-0cf41fd0b9103ec6","claim":"hadamard_kernel"},{"node_id":"n-0cf41fd0b9103ec6","claim":"real_rooted"},{"node_id":"n-0cf41fd0b9103ec6","claim":"interlacing"}]}