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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":"a095dc191c87dcf0bcc8686f15ccbcefabd1a946f272d65d4b9c9c1dc34c6391","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-47e3a1a59edf16da","role":"sequence","name":"A triangle of numbers related to triangle A030528; array a(n,m), read by rows (1 <= m <= n).","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: 3,  5,   1;\n  1: 3, 15,   9,    1;\n  2: 0, 30,  45,   14,     1;\n  3: 0, 30, 150,  105,    20,      1;\n  4: 0,  0, 315,  525,   210,     27,      1;\n  5: 0,  0, 315, 1785,  1470,    378,     35,      1;\n  6: 0,  0,   0, 3780,  7245,   3528,    630,     44,     1;\n  7: 0,  0,   0, 3780, 24570,  23625,   7560,    990,    54,    1;\n  8: 0,  0,   0,    0, 51975, 114345,  65835,  14850,  1485,   65,  1;\n  9: 0,  0,   0,    0, 51975, 384615, 426195, 162855, 27225, 2145, 77, 1;\n","recurrence_sentence":"The row polynomials satisfy (n^2+n+(2*n-4)*x+2*x^2)*P(n,x) = -(n^3-n^2-2*n+(n^2-5*n+6)*x+2*x^2-2*x^3)*P(n-1,x) + ((2*n^2+2*n)*x+(4*n-4)*x^2+4*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 3 + 5*x + x^2.","mathematica":"P[0, x_] := 3 + 5*x + x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - n^2 - 2*n + (n^2 - 5*n + 6)*x + 2*x^2 - 2*x^3)*P[n - 1, x] + ((2*n^2 + 2*n)*x + (4*n - 4)*x^2 + 4*x^3)*D[P[n - 1, x], x])/(n^2 + n + (2*n - 4)*x + 2*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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