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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":"02de9644f0348b4e5ca79c401802ce3881416ef3e8919dbc27d25d1eea65c832","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-c72653d9ab0ee8be","role":"sequence","name":"Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:   0;\n   2:   1;\n   3:   3,    2;\n   4:   7,   10,    3;\n   5:  15,   32,   21,    4;\n   6:  30,   86,   87,   36,    5;\n   7:  58,  208,  285,  184,   55,    6;\n   8: 109,  470,  813,  716,  335,   78,   7;\n   9: 201, 1012, 2115, 2364, 1515,  552, 105,   8;\n  10: 365, 2102, 5151, 6984, 5700, 2850, 847, 136, 9;\n","recurrence_sentence":"The row polynomials satisfy (n^2-4*n+4-(n-2)*x)*P(n,x) = -(2*n^2-17*n+30-(5*n-16)*x+2*x^2)*P(n-1,x) + (5*n-15+(6*n-23)*x+(n-9)*x^2-x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 17*n + 30 - (5*n - 16)*x + 2*x^2)*P[n - 1, x] + (5*n - 15 + (6*n - 23)*x + (n - 9)*x^2 - x^3)*D[P[n - 1, x], x])/(n^2 - 4*n + 4 - (n - 2)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); 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