{"id":"e-952f8a74f60008bb","operation":{"operation":"theorem_check","theorem":"liu_wang_derivative_lag"},"operation_detail":"liu_wang_derivative_lag","conclusion_id":"n-71548025909043e7","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":"sign","lean_declarations":["RealRooted.threeTerm_interlaces_of_nonnegCoeffs","RealRooted.LiuWang.strictInterl_nonpos_lag_sequence"],"certificate_hash":"da29681ac44f2fbac52ea27fe338d44b1ad81d62d5bc8d6a8fefb4d3e68308ee","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":5907,"certificate_compressed_size":1607,"conclusion_name":"Triangle T(n,k,p,q,j) = T(n-1,k,p,q,j) + T(n-1,k-1,p,q,j) + (p*j+q)*prime(j)*T(n-2,k-1,p,q,j) with T(2,k,p,q,j) = prime(j) and (p,q,j) = (0,1,2), read by rows.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"strict","orientation":"previous_interlaces_next"}],"premises":[],"cited_by":[{"node_id":"n-71548025909043e7","claim":"real_rooted"},{"node_id":"n-71548025909043e7","claim":"one_signed_coefficients"},{"node_id":"n-71548025909043e7","claim":"hadamard_kernel"},{"node_id":"n-71548025909043e7","claim":"interlacing"}]}