{"id":"e-1e84ca509450e65e","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-fb40e29cda26e7b9","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":"bfbae237020996873f39a9f831f6b75e926b7ae6da1b7d3d1a2cd51b5f23d9db","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":15371,"certificate_compressed_size":2966,"conclusion_name":"one_plus_derivative of Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1, k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles.","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-125f84bee6f03b65","role":"sequence","name":"one_plus_derivative of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= 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-fb40e29cda26e7b9","claim":"real_rooted"},{"node_id":"n-fb40e29cda26e7b9","claim":"one_signed_coefficients"},{"node_id":"n-fb40e29cda26e7b9","claim":"hadamard_kernel"},{"node_id":"n-fb40e29cda26e7b9","claim":"interlacing"}]}