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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":"32c71073e96633a3ca4b3d625eb9ad3c3488e4197900206362c9528d9f70c1d4","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-fabb0a68e001a863","role":"sequence","name":"Triangle of cubed binomial coefficients, T(n,k) = C(n,k)^3, read by rows.","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:   0;\n  1:   1;\n  2:   8,     2;\n  3:  27,    54,       3;\n  4:  64,   432,     192,       4;\n  5: 125,  2000,    3000,     500,        5;\n  6: 216,  6750,   24000,   13500,     1080,       6;\n  7: 343, 18522,  128625,  171500,    46305,    2058,      7;\n  8: 512, 43904,  526848, 1372000,   878080,  131712,   3584,    8;\n  9: 729, 93312, 1778112, 8001504, 10001880, 3556224, 326592, 5832, 9;\n","recurrence_sentence":"The row polynomials satisfy (n^4-3*n^3+3*n^2-n)*P(n,x) = (n^4+(10*n^4-45*n^3+72*n^2-48*n+12)*x)*P(n-1,x) + ((3*n^3+9*n^2-15*n+6)*x-(15*n^3-51*n^2+48*n-15)*x^2)*d/dx P(n-1,x) + ((6*n^2-8*n+3)*x^2+(6*n^2-8*n+3)*x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n^4 + (10*n^4 - 45*n^3 + 72*n^2 - 48*n + 12)*x)*P[n - 1, x] + ((3*n^3 + 9*n^2 - 15*n + 6)*x - (15*n^3 - 51*n^2 + 48*n - 15)*x^2)*D[P[n - 1, x], x] + ((6*n^2 - 8*n + 3)*x^2 + (6*n^2 - 8*n + 3)*x^3)*D[P[n - 1, x], {x, 2}])/(n^4 - 3*n^3 + 3*n^2 - n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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