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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":"ae055cffa496b71da5f8036322f1edcb939ac2788496cfea43463fbc695b77d7","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-30cb3cad216ed189","role":"sequence","name":"Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).","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:     23,      10,       1;\n  3:    102,      70,      15,      1;\n  4:    441,     418,     141,     20,      1;\n  5:   1871,    2275,    1060,    236,     25,     1;\n  6:   7823,   11644,    6966,   2140,    355,    30,    1;\n  7:  32332,   57036,   41814,  16490,   3770,   498,   35,   1;\n  8: 132369,  270300,  235110, 113644,  33335,  6062,  665,  40,  1;\n  9: 537689, 1248357, 1258164, 721820, 259469, 60501, 9128, 856, 45, 1;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3-2*n-10*n*x-2*n*x^2)*P(n,x) = (4*n^3+12*n^2-4*n-12+(n^3+3*n^2-26*n-28)*x-(10*n+10)*x^2-(n+1)*x^3)*P(n-1,x) + (12*n^2+12*n-96+(8*n^2+8*n-148)*x+(n^2+n-76)*x^2-15*x^3-x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n^3 + 12*n^2 - 4*n - 12 + (n^3 + 3*n^2 - 26*n - 28)*x - (10*n + 10)*x^2 - (n + 1)*x^3)*P[n - 1, x] + (12*n^2 + 12*n - 96 + (8*n^2 + 8*n - 148)*x + (n^2 + n - 76)*x^2 - 15*x^3 - x^4)*D[P[n - 1, x], x])/(2*n^3 - 2*n - 10*n*x - 2*n*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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