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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":"7eb63b97decf05acee3835fa16f9a5aaaa9bb2f35edf14774707f91d3d8434f7","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-67a75673619cc276","role":"sequence","name":"Square array read by antidiagonals downwards: T(n,k), n>=0, k>=0, in which column 0 is equal to A057427: 0, 1, 1, 1, ..., and for k > 0 column k lists two zeros followed by the partial sums of column k-1.","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,   1;\n  1:  1,   2,   1;\n  2:  1,   3,   3,    1;\n  3:  2,   7,   9,    5,    1;\n  4:  3,  13,  22,   18,    7,    1;\n  5:  4,  21,  45,   50,   30,    9,    1;\n  6:  6,  36,  91,  125,  100,   46,   11,   1;\n  7:  9,  61, 178,  291,  290,  179,   66,  13,   1;\n  8: 13,  99, 330,  630,  756,  588,  294,  90,  15,  1;\n  9: 19, 160, 598, 1303, 1827, 1715, 1085, 453, 118, 17, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+2*x+x^2)*P(n-3,x) + (1+x)*P(n-1,x) for n >= 3, with P(0,x) = 1 + x, P(1,x) = 1 + 2*x + x^2, P(2,x) = 1 + 3*x + 3*x^2 + x^3.","mathematica":"P[0, x_] := 1 + x;\nP[1, x_] := 1 + 2*x + x^2;\nP[2, x_] := 1 + 3*x + 3*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[(1 + 2*x + x^2)*P[n - 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