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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":"9ec55f82dac50f936421d81f3642098ce748aa0f6163706e1806c4c70914ee18","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-e3a9cbbad5cb7ab4","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of words over alphabet {0,1,2,3} having exactly k occurrences of the string 01, where n>=0 and k>=0.","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:      1;\n  1:      4;\n  2:     16,      1;\n  3:     64,      8;\n  4:    255,     48,     1;\n  5:   1012,    256,    12;\n  6:   4002,   1277,    96,    1;\n  7:  15776,   6096,   640,   16;\n  8:  62013,  28200,  3834,  160,  1;\n  9: 243140, 127392, 21384, 1280, 20;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3+18*n^2-32*n+(8*n^2-4*n)*x+8*n*x^2)*P(n,x) = (7*n^3+72*n^2-121*n-6+(n^3+36*n^2-29*n+8)*x+(4*n^2+26*n+2)*x^2+(4*n-4)*x^3)*P(n-1,x) + (6*n^2+66*n-60-(4*n^2+20*n-76)*x-(2*n^2+38*n-20)*x^2-(8*n+28)*x^3-8*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((7*n^3 + 72*n^2 - 121*n - 6 + (n^3 + 36*n^2 - 29*n + 8)*x + (4*n^2 + 26*n + 2)*x^2 + (4*n - 4)*x^3)*P[n - 1, x] + (6*n^2 + 66*n - 60 - (4*n^2 + 20*n - 76)*x - (2*n^2 + 38*n - 20)*x^2 - (8*n + 28)*x^3 - 8*x^4)*D[P[n - 1, x], x])/(2*n^3 + 18*n^2 - 32*n + (8*n^2 - 4*n)*x + 8*n*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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