{"id":"e-1b432492072e2170","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-6739edea1e96ea71","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":"7f48214f4745d3498fb76f5e85d8c569181ab239f9033480f84e310a708c48ee","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":12524,"certificate_compressed_size":2186,"conclusion_name":"shift_x_plus_one of shift_x_plus_one of Number triangle T(n,k)=C(n,k)C(2,n-k).","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-5a583a8eb9880032","role":"sequence","name":"shift_x_plus_one of Number triangle T(n,k)=C(n,k)C(2,n-k).","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-6739edea1e96ea71","claim":"one_signed_coefficients"},{"node_id":"n-6739edea1e96ea71","claim":"hadamard_kernel"},{"node_id":"n-6739edea1e96ea71","claim":"real_rooted"}]}