{"id":"e-cd198aa968243143","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-ff2ca64442a7da44","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":"0359ca3b2d75a2ee7d876ec45c953d5ce79391cb710b5c5ba7a85bc43e686668","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":12559,"certificate_compressed_size":2204,"conclusion_name":"shift_x_plus_one of shift_x_plus_one of Triangle read by rows: T(n,k) = k*(k+1)*binomial(n,k)/2 (1 <= k <= n).","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-ff3909e492eb580d","role":"sequence","name":"shift_x_plus_one of Triangle read by rows: T(n,k) = k*(k+1)*binomial(n,k)/2 (1 <= 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-ff2ca64442a7da44","claim":"one_signed_coefficients"},{"node_id":"n-ff2ca64442a7da44","claim":"hadamard_kernel"},{"node_id":"n-ff2ca64442a7da44","claim":"real_rooted"}]}