{"id":"e-01f9ee4e03bf1a78","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-1d2f6070a65c27f0","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":"bffc467011a746c6bad778bfb17af633f208ee2ed5ebb2cf05fa6ad6e4b9471c","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":10663,"certificate_compressed_size":2354,"conclusion_name":"basis_map_A013609 of Triangle read by rows: The convolution triangle of 2^n, where the convolution triangle of a sequence is defined in A357368.","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-19598058c018a2ec","role":"sequence","name":"Triangle whose rows are generated by A136157^n * [1, 1, 0, 0, 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-1d2f6070a65c27f0","claim":"real_rooted"},{"node_id":"n-1d2f6070a65c27f0","claim":"one_signed_coefficients"},{"node_id":"n-1d2f6070a65c27f0","claim":"hadamard_kernel"},{"node_id":"n-1d2f6070a65c27f0","claim":"interlacing"}]}