{"id":"e-69945ed2f7dd729f","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-af416c91e2aef37c","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":"99c3eee2960936c28224d62940be378401118293515dc648fefd3baf98583757","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":12495,"certificate_compressed_size":2182,"conclusion_name":"shift_x_plus_one of Triangle read by rows: T(n,k) = ((n+1)(n+2)/2) * binomial(n,k) (0 <= k <= n).","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-7f1b09730105c8f0","role":"sequence","name":"Triangle read by rows: T(n,k) = ((n+1)(n+2)/2) * binomial(n,k) (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-af416c91e2aef37c","claim":"one_signed_coefficients"},{"node_id":"n-af416c91e2aef37c","claim":"hadamard_kernel"},{"node_id":"n-af416c91e2aef37c","claim":"real_rooted"}]}