{"id":"e-6e3ca58296ce0220","operation":{"operation":"theorem_check","theorem":"ma_wang_interval"},"operation_detail":"ma_wang_interval","conclusion_id":"n-5b857304d1bafe7a","premise_count":0,"theorem":"S.-M. Ma and Y. Wang, q-Eulerian polynomials and polynomials with only real zeros, Electron. J. Combin. 15 (2008) R17, Theorem 2 and Corollary 3 (arXiv:math/0611826), in the closed two-endpoint form of the RealRooted library (https://github.com/PerAlexandersson/RealRooted, RealRooted.affineEndpointStep_closed), after the change of variable x = U + (U - L) y, with the endpoint multiplicities tracked exactly through the induction","lean_route":null,"lean_declarations":["RealRooted.affineEndpointStep_closed","RealRooted.affineEndpointStep_roots_mem_Icc_iff","RealRooted.affineEndpointStep_rootMultiplicity_zero","RealRooted.affineEndpointStep_rootMultiplicity_neg_one","RealRooted.MaWang.endpointStep","RealRooted.MaWang.affineEndpointSequence","RealRooted.normalizedAffineEndpointSequence_roots_mem_Icc_iff"],"certificate_hash":"1a12b5a298772faba998e293a26445df4233899555ac26e8233fdc10ec191fb2","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":4022,"certificate_compressed_size":1337,"conclusion_name":"Triangle read by rows, T(n, k) = Sum_{j=0..n} C(n-j, n-k)*E2(n, j), where E2 are the second-order Eulerian numbers A201637, for n >= 0 and 0 <= k <= n.","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[],"cited_by":[{"node_id":"n-5b857304d1bafe7a","claim":"real_rooted"},{"node_id":"n-5b857304d1bafe7a","claim":"one_signed_coefficients"},{"node_id":"n-5b857304d1bafe7a","claim":"hadamard_kernel"},{"node_id":"n-5b857304d1bafe7a","claim":"interlacing"}]}