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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":"525b385d4e45f2d72607d94792b6fe0784dc243179cc5423882fb681be8dc73f","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-eacb751f39dbd056","role":"sequence","name":"Expansion of 1/sqrt(1-4*x*y-4*x^2*y).","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: 2,  2;\n  2: 2, 14,   6;\n  3: 0, 24,  72,   20;\n  4: 0, 12, 186,  340,    70;\n  5: 0,  0, 180, 1180,  1540,    252;\n  6: 0,  0,  60, 1700,  6720,   6804,    924;\n  7: 0,  0,   0, 1120, 12880,  35784,  29568,   3432;\n  8: 0,  0,   0,  280, 12670,  85680, 182028, 126984,  12870;\n  9: 0,  0,   0,    0,  6300, 112140, 522984, 895752, 540540, 48620;\n","recurrence_sentence":"The row polynomials satisfy (n^3-n^2+(2*n^2-4*n)*x+(4*n^2-6*n)*x^2+4*n*x^3)*P(n,x) = -(2*n^3-8*n^2+6*n-(6*n^2+2*n-12)*x+(8*n^2-16*n+16)*x^2-4*x^3-8*x^4)*P(n-1,x) + ((4*n^2-4*n)*x+(4*n^2+4*n-8)*x^2+24*n*x^3+(16*n+24)*x^4+16*x^5)*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[(-(2*n^3 - 8*n^2 + 6*n - (6*n^2 + 2*n - 12)*x + (8*n^2 - 16*n + 16)*x^2 - 4*x^3 - 8*x^4)*P[n - 1, x] + ((4*n^2 - 4*n)*x + (4*n^2 + 4*n - 8)*x^2 + 24*n*x^3 + (16*n + 24)*x^4 + 16*x^5)*D[P[n - 1, x], x])/(n^3 - n^2 + (2*n^2 - 4*n)*x + (4*n^2 - 6*n)*x^2 + 4*n*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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