{"id":"e-c3e188a1b6271e04","operation":{"operation":"theorem_check","theorem":"iterated_preserver"},"operation_detail":"iterated_preserver","conclusion_id":"n-380b79aacd7d0dd2","premise_count":0,"theorem":"Iterated real-rootedness preservers (induction on n): if P_m is nonzero and real-rooted with every root in a closed region R and, for every n > m, P_n = a_n(x) S_n(P_(n-1)) where S_n maps every polynomial that is zero or real-rooted with every root in R to zero or a real-rooted polynomial with every root in R and a_n is real-rooted with every root in R, then every P_n is real-rooted with every root in R (the roots of a product are the roots of its factors); an exact degree step keeps every row nonzero; here S_n is the identity; for R = [0, inf) the reflection x -> -x maps the rows to rows with every root in (-inf, 0]","lean_route":null,"lean_declarations":[],"certificate_hash":"11dcf6d40eec87c72bcf9772b7c9a17e0a71f85be3ea92bb81f528403465bc03","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":3850,"certificate_compressed_size":1336,"conclusion_name":"Triangle read by rows: T(n, m) = S1(n, m)*6^(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":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[],"cited_by":[{"node_id":"n-380b79aacd7d0dd2","claim":"real_rooted"},{"node_id":"n-380b79aacd7d0dd2","claim":"one_signed_coefficients"},{"node_id":"n-380b79aacd7d0dd2","claim":"interlacing"}]}