{"id":"e-5f21311bb67f8c1c","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-efe23a76953cf69d","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":"a0ec723de98c4f166dadb3dcf8b252126ac491e89e8ba5e1c34fe8e99864a5b4","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":2735,"certificate_compressed_size":1031,"conclusion_name":"Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1, k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[],"cited_by":[{"node_id":"n-efe23a76953cf69d","claim":"real_rooted"},{"node_id":"n-efe23a76953cf69d","claim":"one_signed_coefficients"},{"node_id":"n-efe23a76953cf69d","claim":"hadamard_kernel"},{"node_id":"n-efe23a76953cf69d","claim":"interlacing"}]}