{"id":"e-f17aff623964d25a","operation":{"operation":"parent_provers","route":"root_domain"},"operation_detail":"root_domain","conclusion_id":"n-914700b1060feeb2","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":"1d6fbe490426718de898b8aa2ebe3784acfb249d2ee764133a045e8e4a8a81a1","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"parent_provers","certificate_raw_size":1149,"certificate_compressed_size":606,"conclusion_name":"Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirling numbers of the first kind (n >= 2; 1 <= k <= n-1; s(n,j) = A008275(n,j)).","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[],"cited_by":[{"node_id":"n-914700b1060feeb2","claim":"one_signed_coefficients"},{"node_id":"n-914700b1060feeb2","claim":"hadamard_kernel"},{"node_id":"n-914700b1060feeb2","claim":"real_rooted"}]}