{"id":"e-b29f35705824bef6","operation":{"operation":"parent_provers","route":"paper_theorem"},"operation_detail":"paper_theorem","conclusion_id":"n-4d60c3dd8627c01d","premise_count":0,"theorem":"polytool parent provers, route paper_theorem: every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign","lean_route":null,"lean_declarations":[],"certificate_hash":"4dd6f46ba6c2bbe56e3cf1b985c1b401e3e63c8a5eb86ee6f135f8bdf664a0b7","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"parent_provers","certificate_raw_size":1980,"certificate_compressed_size":852,"conclusion_name":"Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first kind in reverse order).","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[],"cited_by":[{"node_id":"n-4d60c3dd8627c01d","claim":"one_signed_coefficients"},{"node_id":"n-4d60c3dd8627c01d","claim":"hadamard_kernel"},{"node_id":"n-4d60c3dd8627c01d","claim":"interlacing"},{"node_id":"n-4d60c3dd8627c01d","claim":"real_rooted"}]}