{"id":"e-62efc507a1cd208a","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-a2304b34e998691c","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 g = f' (as in their Corollary 2.4), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","lean_route":null,"lean_declarations":[],"certificate_hash":"9fcbb9857a62c834d14d0049f06e71cca7106a276b2f9c9972757cdeee66bf17","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":4322,"certificate_compressed_size":1313,"conclusion_name":"Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n - k + c, k) * z^k / k! and c = 4.","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[],"cited_by":[{"node_id":"n-a2304b34e998691c","claim":"interlacing"},{"node_id":"n-a2304b34e998691c","claim":"real_rooted"},{"node_id":"n-a2304b34e998691c","claim":"one_signed_coefficients"},{"node_id":"n-a2304b34e998691c","claim":"hadamard_kernel"}]}