{"id":"e-76296f2a906d8153","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-29982f17f6b1d2a0","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":"1c31d0413450050ca610638cce12eef1bbad750f953854cb287740ea2c29d93f","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":13624,"certificate_compressed_size":2605,"conclusion_name":"basis_map_A013610 of basis_map_A013611 of Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the Formula section.","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-b85d9f7216f04287","role":"sequence","name":"basis_map_A013611 of Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the Formula section.","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-29982f17f6b1d2a0","claim":"real_rooted"},{"node_id":"n-29982f17f6b1d2a0","claim":"one_signed_coefficients"},{"node_id":"n-29982f17f6b1d2a0","claim":"hadamard_kernel"},{"node_id":"n-29982f17f6b1d2a0","claim":"interlacing"}]}