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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":"36f1a02e818284d0bfab80f6434683cf8d2194bff87645af572578ea0cd1dad9","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-688f70ea6f82c256","role":"sequence","name":"Stirling2 triangle with scaled diagonals (powers of 8).","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:             89,             26,              1;\n  1:           1009,            547,             51,              1;\n  2:          13457,          11124,           1846,             84,             1;\n  3:         210105,         232101,          59290,           4650,           125,            1;\n  4:        3747753,        5104462,        1825951,         216740,          9815,          174,           1;\n  5:       74565473,      119738823,       55954997,        9292451,        626115,        18389,         231,         1;\n  6:     1628999841,     3005405416,     1741632572,      382641560,      35698726,      1538264,       31612,       296,        1;\n  7:    38704241897,    80587013065,    55729617156,    15511163284,    1907819886,    112591038,     3356724,     50916,      369,      1;\n  8:   993034281593,  2300357338498,  1846095360781,   628648241880,   98235420130,   7585903500,   306775602,   6689880,    77925,    450,   1;\n  9: 27340167242321, 69636634808555, 63540299163143, 25735020786981, 4959736607210, 484669920526, 25446746094, 746605002, 12410805, 114455, 539, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (9+x)*P(n-1,x) + (8+8*x)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 89 + 26*x + x^2.","mathematica":"P[0, x_] := 89 + 26*x + x^2;\nP[n_, x_] := P[n, x] = Expand[(9 + x)*P[n - 1, x] + (8 + 8*x)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 89 + 26*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (9 + x)*v[i - 1] + (8 + 8*x)*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-688f70ea6f82c256","steps":["x \u21a6 x+1"],"edges":["e-6a225eefeba308c6"],"text":"A075503 \u2192 x \u21a6 x+1"},"headline":"A075503 \u2192 x \u21a6 x+1","chain":null}