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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":"5b74293e238715a9e857d4480261e1f40dba5f6c4b5edbc67f65b18c6bd0a52c","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-964d853754e5a7e8","role":"sequence","name":"reversal of A133080 * A007318.","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:  2;\n  2:  2,  2;\n  3:  4, 10,   6;\n  4:  4, 12,  12,   4;\n  5:  6, 28,  48,  36,  10;\n  6:  6, 30,  60,  60,  30,   6;\n  7:  8, 54, 150, 220, 180,  78,  14;\n  8:  8, 56, 168, 280, 280, 168,  56,   8;\n  9: 10, 88, 336, 728, 980, 840, 448, 136, 18;\n","recurrence_sentence":"The row polynomials satisfy (n^2-3*n+2)*P(n,x) = (3*n^2-5*n-2+(2*n^3-10*n+4)*x+(3*n^3-9*n^2+6*n)*x^2)*P(n-1,x) - (2*n+(2*n^2+4*n-2)*x+(5*n^2-n-2)*x^2+(3*n^2-3*n)*x^3)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 2, P(2,x) = 2 + 2*x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[2, x_] := 2 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((3*n^2 - 5*n - 2 + (2*n^3 - 10*n + 4)*x + (3*n^3 - 9*n^2 + 6*n)*x^2)*P[n - 1, x] - (2*n + (2*n^2 + 4*n - 2)*x + (5*n^2 - n - 2)*x^2 + (3*n^2 - 3*n)*x^3)*D[P[n - 1, x], x])/(n^2 - 3*n + 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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