{"id":"e-e55b55654f8f2c51","operation":{"operation":"theorem_check","theorem":"liu_wang_three_term"},"operation_detail":"liu_wang_three_term","conclusion_id":"n-cd3918419c5686a8","premise_count":0,"theorem":"L. L. Liu and Y. Wang, Adv. Appl. Math. 38 (2007) 542-560, Theorem 2.1 and Corollary 2.4 (A unified approach to polynomial sequences with only real zeros; the lag row as the interlacer, b_n = 0), weak interlacing of consecutive rows, with nonnegativity of the coefficients replaced, where a root region is certified, by an exact root-location induction","lean_route":"sign","lean_declarations":["RealRooted.threeTerm_interlaces_of_eval_nonpos","RealRooted.threeTerm_interlaces","RealRooted.eval_nonpos_of_natDegree_le_two"],"certificate_hash":"b2fe08292f7d6eb2f6244498f33e5068067ba37931a34ed8df68165057d87469","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":4861,"certificate_compressed_size":1641,"conclusion_name":"Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j,-n)*S1(j,k), S1 the Stirling cycle numbers A132393, for n>=0 and 0<=k<=n.","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[],"cited_by":[{"node_id":"n-cd3918419c5686a8","claim":"real_rooted"},{"node_id":"n-cd3918419c5686a8","claim":"one_signed_coefficients"},{"node_id":"n-cd3918419c5686a8","claim":"hadamard_kernel"},{"node_id":"n-cd3918419c5686a8","claim":"interlacing"}]}