{"id":"e-6629899d205ea17b","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-87f3394ac59b064b","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":"c4fd82883b459db51bedaffe392b3e1c2a9d0efc0d031c0c8032d1ceb68ea296","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":10546,"certificate_compressed_size":2314,"conclusion_name":"basis_map_A007318 of Triangle read by rows: row n consists of the nonzero coefficients of the expansion of 2^n x^n in terms of Hermite polynomials with decreasing subscripts.","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-c914b69838ac70ec","role":"sequence","name":"Triangle read by rows: row n consists of the nonzero coefficients of the expansion of 2^n x^n in terms of Hermite polynomials with decreasing subscripts.","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-87f3394ac59b064b","claim":"real_rooted"},{"node_id":"n-87f3394ac59b064b","claim":"one_signed_coefficients"},{"node_id":"n-87f3394ac59b064b","claim":"hadamard_kernel"},{"node_id":"n-87f3394ac59b064b","claim":"interlacing"}]}