{"id":"e-461c5fccfb2b2d54","operation":{"operation":"theorem_check","theorem":"liu_wang_three_term"},"operation_detail":"liu_wang_three_term","conclusion_id":"n-71e25d00553f2c01","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"],"certificate_hash":"dd09368e03937e27ed495fc1c371314fc275dc17485410ecebb0a0e673b66fae","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":4942,"certificate_compressed_size":1555,"conclusion_name":"Triangle T(n, k, j) =  T(n-1, k, j) + T(n-1, k-1, j) + (2*j + 1)*prime(j)*T(n-2, k-1, j) with T(2, k, j) = prime(j) and j = 8, read by rows.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[],"cited_by":[{"node_id":"n-71e25d00553f2c01","claim":"real_rooted"},{"node_id":"n-71e25d00553f2c01","claim":"one_signed_coefficients"},{"node_id":"n-71e25d00553f2c01","claim":"hadamard_kernel"},{"node_id":"n-71e25d00553f2c01","claim":"interlacing"}]}