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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":"eff8708a57ecdcac8e318353929512237b6b0247049fc3af40bec6c470c4ca8b","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-b1a7dcc8610d7e32","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of ternary strings of length n with k indispensable digits, with 0 <= k <= n.","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,    6;\n  3:  9,   26,   12;\n  4: 14,   70,   78,    20;\n  5: 20,  150,  288,   180,    30;\n  6: 27,  280,  801,   864,   355,    42;\n  7: 35,  476, 1869,  3000,  2135,   630,   56;\n  8: 44,  756, 3864,  8492,  9000,  4620, 1036,   72;\n  9: 54, 1140, 7308, 20844, 30230, 23148, 9072, 1608, 90;\n","recurrence_sentence":"The row polynomials satisfy 4*P(n,x) = (8+6*x-6*x^2+4*x^3)*P(n-3,x) + (1-3*x-3*x^2-4*x^3)*d/dx P(n-3,x) - (20+8*x+12*x^2)*P(n-2,x) - (2-6*x-8*x^2)*d/dx P(n-2,x) + (14+12*x)*P(n-1,x) + (1-4*x)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 2, P(2,x) = 5 + 6*x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[2, x_] := 5 + 6*x;\nP[n_, x_] := P[n, x] = Expand[Together[((8 + 6*x - 6*x^2 + 4*x^3)*P[n - 3, x] + (1 - 3*x - 3*x^2 - 4*x^3)*D[P[n - 3, x], x] - (20 + 8*x + 12*x^2)*P[n - 2, x] - (2 - 6*x - 8*x^2)*D[P[n - 2, x], x] + (14 + 12*x)*P[n - 1, x] + (1 - 4*x)*D[P[n - 1, x], x])/(4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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