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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":"2158765441b6679fd7f074c843cea45716098f4408adaae201b72881a59c1baf","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-7ac21ea1b3b7e282","role":"sequence","name":"Pascal-(1,9,1) array.","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:  1;\n  2: 11,    2;\n  3: 21,   42,     3;\n  4: 31,  282,    93,      4;\n  5: 41,  722,  1083,    164,      5;\n  6: 51, 1362,  5973,   2724,    255,      6;\n  7: 61, 2202, 17763,  23684,   5505,    366,     7;\n  8: 71, 3242, 39453, 119044,  65755,   9726,   497,   8;\n  9: 81, 4482, 74043, 384804, 481005, 148086, 15687, 648, 9;\n","recurrence_sentence":"The row polynomials satisfy -(10*n^2-29*n+19-(10*n^2-8*n-2)*x+(n-1)*x^2)*P(n,x) = -(10*n^2-19*n+9+(190*n^2-752*n+504)*x-(49*n-65)*x^2+2*x^3)*P(n-1,x) + ((10*n-9)*x+(380*n-343)*x^2+(10*n-47)*x^3-x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(10*n^2 - 19*n + 9 + (190*n^2 - 752*n + 504)*x - (49*n - 65)*x^2 + 2*x^3)*P[n - 1, x] + ((10*n - 9)*x + (380*n - 343)*x^2 + (10*n - 47)*x^3 - x^4)*D[P[n - 1, x], x])/(-10*n^2 + 29*n - 19 + (10*n^2 - 8*n - 2)*x - (n - 1)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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