{"id":"e-41f01d4be10aeae3","operation":{"index_shift":2,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"-1/2\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 2","conclusion_id":"n-552ecb7379159c73","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"8b0947d552d466d5d99a2dbab78fd3358ba9050238c6d7e95ed255f2b99ce703","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"preserver_image","certificate_raw_size":10426,"certificate_compressed_size":2942,"conclusion_name":"Triangle read by rows: T(n, m) = S1(n, m)*4^(n-m), where S1 are the signed Stirling numbers of first kind A008275 (n >= 1, 1 <= m <= n).","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"2"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-4f6e747b794bcc2e","role":"sequence","name":"Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.","uses":{"claim":"real_rooted","from_index":2,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 2, is zero or real-rooted with all roots in (-inf, 0]"}],"cited_by":[{"node_id":"n-552ecb7379159c73","claim":"one_signed_coefficients"},{"node_id":"n-552ecb7379159c73","claim":"interlacing"},{"node_id":"n-552ecb7379159c73","claim":"real_rooted"}]}