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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":"d94a7ca63eb3c2855ba82f008f5054edab0ff4837646aa7fae5b44f2ec536135","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-9fa98aec39d32ffa","role":"sequence","name":"Triangle, read by rows, of pairwise sums of trinomial coefficients (A027907).","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:   3,    1;\n  2:   8,    7,     1;\n  3:  17,   31,    12,     1;\n  4:  31,   96,    77,    18,     1;\n  5:  51,  237,   321,   155,    25,     1;\n  6:  78,  503,  1017,   827,   275,    33,    1;\n  7: 113,  959,  2677,  3242,  1813,   448,   42,    1;\n  8: 157, 1688,  6170, 10292,  8563,  3556,  686,   52,  1;\n  9: 211, 2793, 12870, 28042, 32101, 19827, 6426, 1002, 63, 1;\n","recurrence_sentence":"The row polynomials satisfy 4*P(n,x) = (9+3*x-9*x^2)*P(n-3,x) + (1-3*x-3*x^2-4*x^3)*d/dx P(n-3,x) - (22+2*x+4*x^2)*P(n-2,x) - (2-6*x-8*x^2)*d/dx P(n-2,x) + (15+8*x)*P(n-1,x) + (1-4*x)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + x, P(2,x) = 8 + 7*x + x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + x;\nP[2, x_] := 8 + 7*x + x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((9 + 3*x - 9*x^2)*P[n - 3, x] + (1 - 3*x - 3*x^2 - 4*x^3)*D[P[n - 3, x], x] - (22 + 2*x + 4*x^2)*P[n - 2, x] - (2 - 6*x - 8*x^2)*D[P[n - 2, x], x] + (15 + 8*x)*P[n - 1, x] + (1 - 4*x)*D[P[n - 1, x], x])/(4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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