{"id":"e-4245d0a31fdf9485","operation":{"operation":"theorem_check","theorem":"liu_wang_three_term"},"operation_detail":"liu_wang_three_term","conclusion_id":"n-b0fc620949fb5990","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_nonnegCoeffs","RealRooted.threeTerm_interlaces","RealRooted.threeTerm_hasNonnegCoeffs"],"certificate_hash":"947657383cb86cf9853ebba892829c78b2832ade2fa88ea5e50e532bc21b3629","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":4005,"certificate_compressed_size":1397,"conclusion_name":"Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.","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-b0fc620949fb5990","claim":"real_rooted"},{"node_id":"n-b0fc620949fb5990","claim":"one_signed_coefficients"},{"node_id":"n-b0fc620949fb5990","claim":"hadamard_kernel"},{"node_id":"n-b0fc620949fb5990","claim":"interlacing"}]}