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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":"ad0040203cbdc9e67159455040cf22bcd1a267557f9bd163ce805a190bc8a560","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-353fb5a029e4e669","role":"sequence","name":"Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.","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:  2;\n  2:  5,   4;\n  3:  9,  18,    6;\n  4: 14,  50,   39,    8;\n  5: 20, 110,  147,   68,   10;\n  6: 27, 210,  420,  324,  105,   12;\n  7: 35, 364, 1008, 1140,  605,  150,   14;\n  8: 44, 588, 2142, 3300, 2530, 1014,  203,  16;\n  9: 54, 900, 4158, 8316, 8580, 4914, 1575, 264, 18;\n","recurrence_sentence":"The row polynomials satisfy -(2*n^3+3*n^2-3*n-2-(4*n^3-12*n^2+12*n-4)*x)*P(n,x) = -(2*n^3+7*n^2+3*n-(3*n^3-12*n^2-3*n)*x-(2*n^3-2*n^2)*x^2)*P(n-1,x) - ((4*n^2+8*n+12)*x-(7*n^2-10*n-3)*x^2-(2*n^2-2*n)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 + 7*n^2 + 3*n - (3*n^3 - 12*n^2 - 3*n)*x - (2*n^3 - 2*n^2)*x^2)*P[n - 1, x] - ((4*n^2 + 8*n + 12)*x - (7*n^2 - 10*n - 3)*x^2 - (2*n^2 - 2*n)*x^3)*D[P[n - 1, x], x])/(-2*n^3 - 3*n^2 + 3*n + 2 + (4*n^3 - 12*n^2 + 12*n - 4)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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