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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":"bbf3bfcb4ff4898ca5341396769744e333207ac9477c509e0ad09e55fbfef718","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-52650ae6398d7ffe","role":"sequence","name":"Triangle T(n,k) of arctangent numbers: expansion of arctan(x)^n/n!.","uses":{"claim":"real_rooted","from_index":1,"region":{}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)"}]}],"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:        -2,         2,        1;\n   2:        -8,        -8,        3,        1;\n   3:        24,       -40,      -20,        4,         1;\n   4:       184,       184,     -120,      -40,         5,        1;\n   5:      -720,      1568,      784,     -280,       -70,        6,      1;\n   6:     -8448,     -8448,     7392,     2464,      -560,     -112,      7,     1;\n   7:     40320,   -104704,   -52352,    25536,      6384,    -1008,   -168,     8,     1;\n   8:    648576,    648576,  -689280,  -229760,     72240,    14448,  -1680,  -240,     9,    1;\n   9:  -3628800,  10720512,  5360256, -3217280,   -804320,   177408,  29568, -2640,  -330,   10,  1;\n  10: -74972160, -74972160, 91901568, 30633856, -11968000, -2393600, 391776, 55968, -3960, -440, 11, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^4-4*n^3+5*n^2-2*n)*P(n-4,x) + (2*n^2-2*n)*x*P(n-3,x) - (2*n^2+x^2)*P(n-2,x) + 2*x*P(n-1,x) for n >= 5, with P(1,x) = -2 + 2*x + x^2, P(2,x) = -8 - 8*x + 3*x^2 + x^3, P(3,x) = 24 - 40*x - 20*x^2 + 4*x^3 + x^4, P(4,x) = 184 + 184*x - 120*x^2 - 40*x^3 + 5*x^4 + x^5.","mathematica":"P[1, x_] := -2 + 2*x + x^2;\nP[2, x_] := -8 - 8*x + 3*x^2 + x^3;\nP[3, x_] := 24 - 40*x - 20*x^2 + 4*x^3 + x^4;\nP[4, x_] := 184 + 184*x - 120*x^2 - 40*x^3 + 5*x^4 + x^5;\nP[n_, x_] := P[n, x] = Expand[-(n^4 - 4*n^3 + 5*n^2 - 2*n)*P[n - 4, x] + (2*n^2 - 2*n)*x*P[n - 3, x] - (2*n^2 + x^2)*P[n - 2, x] + 2*x*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); 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