{"id":"e-3e082ac2cc139530","operation":{"operation":"parent_provers","route":"root_domain"},"operation_detail":"root_domain","conclusion_id":"n-3b2b9e31bd278a34","premise_count":0,"theorem":"polytool parent provers, route root_domain: every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign","lean_route":null,"lean_declarations":[],"certificate_hash":"010f7ec5709036ef07871822c6586425465ded1a234d826be58545e42bda80e2","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"parent_provers","certificate_raw_size":1590,"certificate_compressed_size":711,"conclusion_name":"T(n, k) = Sum_{j=0..n-k} (-1)^j*binomial(n - k + 1, j)*(n - k + 1 - j)^n. Triangle read by rows, T(n, k) for 1 <= k <= n.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[],"cited_by":[{"node_id":"n-3b2b9e31bd278a34","claim":"one_signed_coefficients"},{"node_id":"n-3b2b9e31bd278a34","claim":"hadamard_kernel"},{"node_id":"n-3b2b9e31bd278a34","claim":"real_rooted"}]}