{"chains":[{"edge":{"id":"e-9163e74e00c6af70","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-4b3a068fa525bce8","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"5a2d8c98f7717ab2d4a794106465164dc6ec19c46b8cc8334b24c8f797883cc0","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-6120ac587df8130e","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs of consecutive 0's (0<=k<=ceiling(n/2)).","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]},"premises":[{"premise":{"position":0,"node_id":"n-6120ac587df8130e","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs of consecutive 0's (0<=k<=ceiling(n/2)).","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"},"chains":[{"edge":{"id":"e-783d9fe6617c7ff0","operation":{"operation":"theorem_check","theorem":"liu_wang_three_term"},"operation_detail":"liu_wang_three_term","conclusion_id":"n-6120ac587df8130e","premise_count":0,"theorem":"L. L. Liu and Y. Wang, Adv. Appl. Math. 38 (2007) 542-560, Theorem 2.1 and Corollary 2.4 (A unified approach to polynomial sequences with only real zeros; the lag row as the interlacer, b_n = 0), weak interlacing of consecutive rows, with nonnegativity of the coefficients replaced, where a root region is certified, by an exact root-location induction","lean_route":null,"lean_declarations":["RealRooted.MaWang.strictInterl_of_interlaces_evalCoeff_nonpos"],"certificate_hash":"64f627b778fbd656aaf8f72208796121eac134e005d558ceec8b22019ba4c7b2","verification_polytool":"replayed","verification_lean":"not_attempted","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":[]},"premises":[]}]}]}],"truncated":false,"expanded":2}