{"id":"e-03bc86f7ebb189b6","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-a0bec09b71e3cb85","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":"d6cfe96fc29254e4c630cf8c47c54f241743f43c1f3a786dd5e8492f8b5508b0","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":10264,"certificate_compressed_size":2292,"conclusion_name":"shift_x_plus_one of basis_map_A013609 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.","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-cf97dd8f766eb0a5","role":"sequence","name":"basis_map_A013609 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.","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-a0bec09b71e3cb85","claim":"one_signed_coefficients"},{"node_id":"n-a0bec09b71e3cb85","claim":"hadamard_kernel"},{"node_id":"n-a0bec09b71e3cb85","claim":"interlacing"},{"node_id":"n-a0bec09b71e3cb85","claim":"real_rooted"}]}