{"id":"e-dff7465450f7ae66","operation":{"operation":"theorem_check","theorem":"liu_wang_derivative_lag"},"operation_detail":"liu_wang_derivative_lag","conclusion_id":"n-5c39e26cb9304567","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":"4446923d0c449f736ad808d3660a400f2732068f9b55e43365df5dd8ffa3b238","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":6173,"certificate_compressed_size":1707,"conclusion_name":"Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+2*x+x^2)/(1-x)^4, read by rows.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":4,"kind":"strict","orientation":"previous_interlaces_next"}],"premises":[],"cited_by":[{"node_id":"n-5c39e26cb9304567","claim":"real_rooted"},{"node_id":"n-5c39e26cb9304567","claim":"one_signed_coefficients"},{"node_id":"n-5c39e26cb9304567","claim":"hadamard_kernel"},{"node_id":"n-5c39e26cb9304567","claim":"interlacing"}]}