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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":"a9f96ad96ed2c9361659e95c39b1ff4ace91bed5e08974f3b8f5ebb16c02162f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}},{"claim":"interlacing","from_index":3,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-afb1686ffa6592c0","role":"sequence","name":"Riordan array ((1-x)/(1+x), x/(1+x)^2).","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:   1;\n   2:  -1,    1;\n   3:  -2,   -2,     1;\n   4:   7,   -3,    -3,     1;\n   5: -14,   24,    -4,    -4,     1;\n   6:  23,  -75,    55,    -5,    -5,     1;\n   7: -34,  174,  -231,   104,    -6,    -6,   1;\n   8:  47, -343,   686,  -546,   175,    -7,  -7,  1;\n   9: -62,  608, -1680,  1968, -1100,   272,  -8, -8,  1;\n  10:  79, -999,  3618, -5742,  4653, -1989, 399, -9, -9, 1;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3-12*n^2+20*n-8+(10*n-20)*x-(2*n-4)*x^2)*P(n,x) = -(2*n^3-8*n^2+6*n+4-(n^3-4*n^2-3*n+14)*x-(6*n-12)*x^2+(n-2)*x^3)*P(n-1,x) - ((4*n^2-12*n+16)*x-(n^2-3*n-8)*x^2-7*x^3+x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = -1 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := -1 + x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 - 8*n^2 + 6*n + 4 - (n^3 - 4*n^2 - 3*n + 14)*x - (6*n - 12)*x^2 + (n - 2)*x^3)*P[n - 1, x] - ((4*n^2 - 12*n + 16)*x - (n^2 - 3*n - 8)*x^2 - 7*x^3 + x^4)*D[P[n - 1, x], x])/(2*n^3 - 12*n^2 + 20*n - 8 + (10*n - 20)*x - (2*n - 4)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); 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