{"id":"e-6a96c1059c15d050","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-f860d7ad52328c75","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":"95ae26dad1532a72b66700652d8412693010b84b4bd1b550f5d56b45c899dde1","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":10248,"certificate_compressed_size":2290,"conclusion_name":"shift_x_plus_one of basis_map_A007318 of Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[x, 0] = 1, p[x, -1] = 0, p[x, 1] = m*x and m = 2. Triangle read by rows, for n >= 0 and 0 <= k <= n.","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-634bc9ca21d13c4b","role":"sequence","name":"basis_map_A007318 of Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirling numbers of the first kind (n >= 2; 1 <= k <= n-1; s(n,j) = A008275(n,j)).","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-f860d7ad52328c75","claim":"one_signed_coefficients"},{"node_id":"n-f860d7ad52328c75","claim":"hadamard_kernel"},{"node_id":"n-f860d7ad52328c75","claim":"interlacing"},{"node_id":"n-f860d7ad52328c75","claim":"real_rooted"}]}