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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":"e24e4dc681d9a9be5445c9326f55ebbd82d1467f92951847e77ee3b3ea6c893e","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:   0;\n  1:   2;\n  2:   5,    8;\n  3:  10,   34,    24;\n  4:  19,  102,   147,     64;\n  5:  36,  268,   588,    516,    160;\n  6:  69,  660,  1950,   2580,   1605,    384;\n  7: 134, 1566,  5820,  10300,   9630,   4614,    896;\n  8: 263, 3626, 16233,  35980,  44905,  32298,  12551,  2048;\n  9: 520, 8248, 43176, 114968, 179480, 172200, 100408, 32776, 4608;\n","recurrence_sentence":"The row polynomials satisfy (n^2-3*n+2)*P(n,x) = (4*n-8-(n^3-6*n^2+7*n+2)*x)*P(n-1,x) + (n-3+(n^2-n-4)*x+(n^2-2*n-1)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 2, P(2,x) = 5 + 8*x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[2, x_] := 5 + 8*x;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n - 8 - (n^3 - 6*n^2 + 7*n + 2)*x)*P[n - 1, x] + (n - 3 + (n^2 - n - 4)*x + (n^2 - 2*n - 1)*x^2)*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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