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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":"1f21da98d7ce92ed5fd5f4c7ac46333cbb0657eb71b38c247efc30401cc73719","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-7e0e90f00bfc1e53","role":"sequence","name":"reversal of Stirling2 triangle with scaled diagonals (powers of 2).","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:  6,    8;\n  3: 12,   56,     24;\n  4: 20,  200,    360,      64;\n  5: 30,  520,   2160,    1984,     160;\n  6: 42, 1120,   8400,   19264,   10080,      384;\n  7: 56, 2128,  25200,  108864,  154560,    48768,     896;\n  8: 72, 3696,  63504,  444864, 1243200,  1161600,  228480,    2048;\n  9: 90, 6000, 141120, 1460928, 6804000, 13096320, 8359680, 1046528, 4608;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = -(n+(4*n^2-8*n)*x+(4*n^3-12*n^2+8*n)*x^2)*P(n-2,x) + (2*n*x^2+(4*n^2-4*n)*x^3)*d/dx P(n-2,x) + (2*n-1+(4*n^2-8*n+4)*x)*P(n-1,x) - (2*n-2)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n + (4*n^2 - 8*n)*x + (4*n^3 - 12*n^2 + 8*n)*x^2)*P[n - 2, x] + (2*n*x^2 + (4*n^2 - 4*n)*x^3)*D[P[n - 2, x], x] + (2*n - 1 + (4*n^2 - 8*n + 4)*x)*P[n - 1, x] - (2*n - 2)*x^2*D[P[n - 1, x], x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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