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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":"077c5a4c3e9eed3eb72ded0c1c913e8bcc456868114f7de219e87b3b3609fbbd","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-cf9f646a76b9c283","role":"sequence","name":"A triangular array related to A077028 and distributing the values of A007582.","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:   3,    2;\n  2:   6,   13,     4;\n  3:  11,   44,    41,      8;\n  4:  20,  121,   198,    113,     16;\n  5:  37,  304,   722,    712,    289,     32;\n  6:  70,  729,  2280,   3230,   2250,    705,     64;\n  7: 135, 1700,  6603,  12240,  12205,   6540,   1665,   128;\n  8: 264, 3889, 18046,  41391,  53900,  41279,  17934,  3841,  256;\n  9: 521, 8768, 47300, 129360, 208222, 208096, 129108, 47120, 8705, 512;\n","recurrence_sentence":"The row polynomials satisfy (n^2-n-1+(2*n-2)*x+x^2)*P(n,x) = (5*n-8-(n^3-4*n^2-2*n+7)*x-(2*n^2-7*n+2)*x^2-(n-3)*x^3)*P(n-1,x) + (n-2+(n^2+n-3)*x+(n^2+2*n-1)*x^2+(2*n+1)*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[((5*n - 8 - (n^3 - 4*n^2 - 2*n + 7)*x - (2*n^2 - 7*n + 2)*x^2 - (n - 3)*x^3)*P[n - 1, x] + (n - 2 + (n^2 + n - 3)*x + (n^2 + 2*n - 1)*x^2 + (2*n + 1)*x^3 + x^4)*D[P[n - 1, x], x])/(n^2 - n - 1 + (2*n - 2)*x + x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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