{"id":"e-78e814d9774691df","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-8addbc07b5fd134f","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":"d5016775691e3532fcc79a6221d087f9c80e1fc2975631b91c573e7bb4bca3bf","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":13430,"certificate_compressed_size":2535,"conclusion_name":"basis_map_A007318 of Triangle read by rows: T(n,k) are the coefficients of Charlier polynomials: A046716 transposed, for 0 <= k <= n.","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-3272d5bc697b9cb4","role":"sequence","name":"Triangle read by rows: T(n,k) are the coefficients of Charlier polynomials: A046716 transposed, for 0 <= k <= n.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}],"cited_by":[{"node_id":"n-8addbc07b5fd134f","claim":"real_rooted"},{"node_id":"n-8addbc07b5fd134f","claim":"one_signed_coefficients"},{"node_id":"n-8addbc07b5fd134f","claim":"hadamard_kernel"},{"node_id":"n-8addbc07b5fd134f","claim":"interlacing"}]}