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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":"382898bc349895ec3d6ca636ee90079098d141772830d74cc5822f49107e7b09","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-3255c4dcfe77aa0b","role":"sequence","name":"basis_map_A013609 of Triangle read by rows, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + (n-1)^2*T(n-1, k), for 0 <= k <= n.","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:           3,            2;\n  2:          12,           14,            4;\n  3:          84,          122,           56,           8;\n  4:        1008,         1632,          916,         208,         16;\n  5:       19152,        33024,        20668,        5784,        720,        32;\n  6:      536256,       962976,       644752,      203288,      31728,      2336,       64;\n  7:    20913984,     38628576,     27071280,     9217736,    1643968,    154560,     7168,     128;\n  8:  1087527168,   2050513920,   1484963712,   533464832,  103921808,  11325056,   681856,   20992,   256;\n  9: 72864320256, 139559486976, 103593596544, 38712071168, 8029690800, 966622368, 68334464, 2770176, 59136, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n^2-2*n+4+2*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(n^2 - 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