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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":"3ba94a656afb3d216d40250d4ca938b3883ec261dc64ca250a8925c76894bae3","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-540fd96c9759fa70","role":"sequence","name":"A skewed version of triangular array A122075.","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: 2, 2;\n  2: 1, 7,  3;\n  3: 0, 6, 18,  5;\n  4: 0, 2, 22, 39,   8;\n  5: 0, 0, 12, 64,  80,  13;\n  6: 0, 0,  3, 49, 162, 156,   21;\n  7: 0, 0,  0, 20, 160, 379,  296,   34;\n  8: 0, 0,  0,  4,  91, 456,  836,  549,   55;\n  9: 0, 0,  0,  0,  30, 330, 1188, 1770, 1002, 89;\n","recurrence_sentence":"The row polynomials satisfy 2*P(n,x) = (2+11*x+23*x^2+23*x^3+5*x^4)*P(n-3,x) + (x+5*x^2+9*x^3+5*x^4)*d/dx P(n-3,x) + (2+9*x+10*x^2+5*x^3)*P(n-2,x) + (x+4*x^2+5*x^3)*d/dx P(n-2,x) + (1+4*x-5*x^2)*P(n-1,x) - (1+4*x+5*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2 + 2*x, P(2,x) = 1 + 7*x + 3*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 2*x;\nP[2, x_] := 1 + 7*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((2 + 11*x + 23*x^2 + 23*x^3 + 5*x^4)*P[n - 3, x] + (x + 5*x^2 + 9*x^3 + 5*x^4)*D[P[n - 3, x], x] + (2 + 9*x + 10*x^2 + 5*x^3)*P[n - 2, x] + (x + 4*x^2 + 5*x^3)*D[P[n - 2, x], x] + (1 + 4*x - 5*x^2)*P[n - 1, x] - (1 + 4*x + 5*x^2)*D[P[n - 1, x], x])/(2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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