{"id":"e-fd63abd21a3410cc","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-01897e65d9bb3867","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":"3267a940244538476f0ed66afca91e0ddf0cb2b2afe9920e95a47121b2f78cea","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"preserver_image","certificate_raw_size":7129,"certificate_compressed_size":1941,"conclusion_name":"Triangle T = A007318*A271703; T(n,m)= Sum_{i=0..n} L'(n,i)*binomial(i,m), m=0..n.","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-9e4edf560ed848c1","role":"sequence","name":"Triangle read by rows: the unsigned Lah numbers T(n, k) = binomial(n-1, k-1)*n!/k! if n > 0 and k > 0, T(n, 0) = 0^n and otherwise 0, for n >= 0 and 0 <= k <= n.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}],"cited_by":[{"node_id":"n-01897e65d9bb3867","claim":"one_signed_coefficients"},{"node_id":"n-01897e65d9bb3867","claim":"hadamard_kernel"},{"node_id":"n-01897e65d9bb3867","claim":"real_rooted"}]}