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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":"27547cc9b2a60ceb7d070fe76d2dde93f9ad7df229d4f027cdb2aec4927ce3a5","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-86fb6463ff8368f2","role":"sequence","name":"Triangle of third-order Eulerian numbers: 3-Stirling permutations enumerated by ascents.","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]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:           1;\n  1:           4,            3;\n  2:          28,           42,           15;\n  3:         280,          630,          456,          105;\n  4:        3640,        10920,        11946,         5610,          945;\n  5:       58240,       218400,       320124,       228228,        78660,        10395;\n  6:     1106560,      4979520,      9154908,      8773632,      4603320,      1246770,       135135;\n  7:    24344320,    127807680,    282688560,    340642260,    240785136,     99451530,     22110480,     2027025;\n  8:   608608000,   3651648000,   9441413520,  13713278040,  12209975316,   6804733320,   2309678190,   434475090,   34459425;\n  9: 17041024000, 115026912000, 340419599520, 578867289000, 622158080544, 437338298628, 200517053220, 57627218880, 9378690300, 654729075;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3*n+1+3*n*x)*P(n-1,x) - 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