{"id":"e-63eede96398ad848","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-e1278fe539e1f3ff","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":"62aaafc338cc4cbc37055a241868990cf1e9fff5ab84380cbbd9d173792a74df","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":10913,"certificate_compressed_size":2380,"conclusion_name":"shift_x_plus_one of reversal of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 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-ab4410127fe29869","role":"sequence","name":"reversal 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-e1278fe539e1f3ff","claim":"one_signed_coefficients"},{"node_id":"n-e1278fe539e1f3ff","claim":"hadamard_kernel"},{"node_id":"n-e1278fe539e1f3ff","claim":"interlacing"},{"node_id":"n-e1278fe539e1f3ff","claim":"real_rooted"}]}