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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":"4fd484a3802a9203d06e3cbfa0d5c9ce29d72e4ce8e489a69dad6d6e32ca0b01","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-bf8bba6616804a77","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:      0;\n  1:      1;\n  2:      3;\n  3:     10,      2;\n  4:     42,     14;\n  5:    216,     90,     3;\n  6:   1320,    624,    39;\n  7:   9360,   4800,   408,    4;\n  8:  75600,  41040,  4200,   84;\n  9: 685440, 388080, 45000, 1300, 5;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -x^2*P(n-4,x) - (2*n-4)*x*P(n-3,x) - (n^2-3*n+2-2*x)*P(n-2,x) + (2*n-2)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 3, P(3,x) = 10 + 2*x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[2, x_] := 3;\nP[3, x_] := 10 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-x^2*P[n - 4, x] - (2*n - 4)*x*P[n - 3, x] - (n^2 - 3*n + 2 - 2*x)*P[n - 2, x] + (2*n - 2)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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