{"id":"e-ba136fbd81c91b01","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-f5badeede90e2ff8","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":"f8e9f2adf1c7e7d120bdf99014a1319db5a3d49df4e76ca3ccc4c151f6fd6567","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":12530,"certificate_compressed_size":2193,"conclusion_name":"shift_x_plus_one of reversal 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-2c11b65ec5b65118","role":"sequence","name":"reversal 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-f5badeede90e2ff8","claim":"one_signed_coefficients"},{"node_id":"n-f5badeede90e2ff8","claim":"hadamard_kernel"},{"node_id":"n-f5badeede90e2ff8","claim":"real_rooted"}]}