{"id":"e-f3672e8db5e390e2","operation":{"operation":"theorem_check","theorem":"liu_wang_derivative_lag"},"operation_detail":"liu_wang_derivative_lag","conclusion_id":"n-b24bb505650436c3","premise_count":0,"theorem":"L. L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Appl. Math. 38 (2007) 542-560, Theorem 2.1 with the interlacers f' and the lag row (strict form, as in the derivative-lag theorems of the RealRooted library, https://github.com/PerAlexandersson/RealRooted), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","lean_route":"halfGrowth","lean_declarations":["RealRooted.derivLag_strictInterl_of_nonnegCoeffs","RealRooted.derivLag_strictInterl_and_noCommonRoot","RealRooted.derivLag_eval_zero_pos"],"certificate_hash":"317ce357deedebb1fe17548769fefdae735d5d4fc9eba016a675b97a6172a134","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":6125,"certificate_compressed_size":1714,"conclusion_name":"Triangle read by rows, T(n,k) = Sum_{j=0..n} C(-j-1,-n-1)*E1(j,k), E1 the Eulerian numbers A173018, for n>=0 and 0<=k<=n.","claims":[{"claim":"real_rooted","from_index":3,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":4,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[],"cited_by":[{"node_id":"n-b24bb505650436c3","claim":"real_rooted"},{"node_id":"n-b24bb505650436c3","claim":"one_signed_coefficients"},{"node_id":"n-b24bb505650436c3","claim":"hadamard_kernel"},{"node_id":"n-b24bb505650436c3","claim":"interlacing"}]}