{"id":"e-5dc0992966a8d995","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-9e788b258df52f7d","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":"adde0b1cc61996222c8f24a1455877020797eaec99d0dfe2bd0791e87881bff6","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":14522,"certificate_compressed_size":2898,"conclusion_name":"shift_x_plus_one of T(n,k) = 2^k*binomial(n,2k+1), where 0 <= k <= floor((n-1)/2), n >= 1.","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-286e213a8791bd45","role":"sequence","name":"T(n,k) = 2^k*binomial(n,2k+1), where 0 <= k <= floor((n-1)/2), n >= 1.","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-9e788b258df52f7d","claim":"one_signed_coefficients"},{"node_id":"n-9e788b258df52f7d","claim":"hadamard_kernel"},{"node_id":"n-9e788b258df52f7d","claim":"real_rooted"},{"node_id":"n-9e788b258df52f7d","claim":"interlacing"}]}