{"id":"e-0b9b63070f7f0adc","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-4e8690098ee9d7b4","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":"7bc26e04c1fe531ed76d37e546cd4a63ee4a882bb0711ddc76d24a727fa44cb8","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":6885,"certificate_compressed_size":1392,"conclusion_name":"shift_x_plus_one of Triangle T(n,k) read by rows: number of size-k antisymmetric subsets of {1,2,...,n}.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-ba3bb18490b15e2e","role":"sequence","name":"Triangle T(n,k) read by rows: number of size-k antisymmetric subsets of {1,2,...,n}.","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-4e8690098ee9d7b4","claim":"hadamard_kernel"},{"node_id":"n-4e8690098ee9d7b4","claim":"real_rooted"},{"node_id":"n-4e8690098ee9d7b4","claim":"one_signed_coefficients"}]}