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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":"b85af9bf8a5d75b92f252099047cca41542f6a5435bded691a6d232e6cd0d6dc","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-d0e888e78d074358","role":"sequence","name":"Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A286718 (|S1hat[3,1]| generalized Stirling 1), for n >= 0.","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]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:            1;\n   2:            3,             2;\n   3:           27,            27,             4;\n   4:          405,           540,           171,             8;\n   5:         8505,         14175,          6885,           951,           16;\n   6:       229635,        459270,        300510,         72198,         4995,           32;\n   7:      7577955,      17681895,      14543550,       5038362,       682479,        25551,          64;\n   8:    295540245,     788107320,     780529365,     354760560,     73848915,      6073056,      129099,       128;\n   9:  13299311025,   39897933075,   46202791635,   26040379365,   7403662035,    994529745,    52070121,    648567,     256;\n  10: 678264862275, 2260882874250, 2996778084300, 2017751766030, 731429268570, 139454339310, 12658141188, 436017978, 3249747, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (6*n-9+(3*n-4)*x)*P(n-1,x) - 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