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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":"22552bc170cd3e2a4d38a81941d489de0c223dcde7ac3aab19d822496389f83e","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-e7331b0d50084c15","role":"sequence","name":"Triangular array:  row n gives the coefficients of the polynomial p(n,x) defined in Comments.","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,   3;\n  2:  7,  18,    6;\n  3: 10,  51,   57,    12;\n  4: 13, 102,  225,   156,    24;\n  5: 16, 171,  591,   795,   396,     48;\n  6: 19, 258, 1236,  2604,  2478,    960,     96;\n  7: 22, 363, 2241,  6582,  9753,   7128,   2256,   192;\n  8: 25, 486, 3687, 14052, 29025,  32808,  19392,  5184,   384;\n  9: 28, 627, 5655, 26661, 71634, 112587, 102264, 50640, 11712, 768;\n","recurrence_sentence":"The row polynomials satisfy -(3*n^3+n^2-2*n-(6*n^3-6*n^2-3*n+9)*x-(12*n^2-22*n+10)*x^2-(8*n^2-12*n-4)*x^3-(8*n-8)*x^4)*P(n,x) = -(3*n^3+4*n^2+n+(12*n^3-21*n-9)*x+(12*n^2-20*n-10)*x^2+(16*n^2+12*n+4)*x^3+(16*n+8)*x^4)*P(n-1,x) + ((3*n^2+4*n+1)*x+(24*n^2+40*n+13)*x^2+(12*n^2+84*n+52)*x^3+(64*n+88)*x^4+(16*n+64)*x^5+16*x^6)*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[(-(3*n^3 + 4*n^2 + n + (12*n^3 - 21*n - 9)*x + (12*n^2 - 20*n - 10)*x^2 + (16*n^2 + 12*n + 4)*x^3 + (16*n + 8)*x^4)*P[n - 1, x] + ((3*n^2 + 4*n + 1)*x + (24*n^2 + 40*n + 13)*x^2 + (12*n^2 + 84*n + 52)*x^3 + (64*n + 88)*x^4 + (16*n + 64)*x^5 + 16*x^6)*D[P[n - 1, x], x])/(-3*n^3 - n^2 + 2*n + (6*n^3 - 6*n^2 - 3*n + 9)*x + (12*n^2 - 22*n + 10)*x^2 + (8*n^2 - 12*n - 4)*x^3 + (8*n - 8)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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