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D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"62d6fd394aa0ba49a487f560a7e57ec4f10cfce62accd6929bd3070c717ea6dc","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-61bf3ec06b6c88be","role":"sequence","name":"basis_map_A007318 of Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","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:            5,            1;\n  2:           44,           14,           1;\n  3:          560,          224,          27,          1;\n  4:         9344,         4320,         680,         44,         1;\n  5:       193024,        98944,       18480,       1600,        65,        1;\n  6:      4755456,      2637824,      556864,      58240,      3220,       90,      1;\n  7:    136101888,     80420864,    18645760,    2234624,    151200,     5824,    119,     1;\n  8:   4437835776,   2763096064,   690532352,   91886592,   7198464,   342720,   9744,   152,   1;\n  9: 162404499456, 105704718336, 28106096640, 4067348480, 356778240, 19826688, 702240, 15360, 189, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(16*n^2-32*n+16+(8*n-8)*x+x^2)*P(n-2,x) + (8*n-4+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 5 + x;\nP[n_, x_] := P[n, x] = Expand[-(16*n^2 - 32*n + 16 + (8*n - 8)*x + x^2)*P[n - 2, x] + (8*n - 4 + 2*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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