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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":"4ef9d1617d780fe53a0d68a9f2c2f097e75825d6e470fe67cf409c06f210d1b9","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-f5a001953ee28e8f","role":"sequence","name":"Riordan array (1, 2+2x).","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: 2,  8;\n  3: 0, 16, 24;\n  4: 0,  8, 72,  64;\n  5: 0,  0, 72, 256,  160;\n  6: 0,  0, 24, 384,  800,  384;\n  7: 0,  0,  0, 256, 1600, 2304,   896;\n  8: 0,  0,  0,  64, 1600, 5760,  6272,  2048;\n  9: 0,  0,  0,   0,  800, 7680, 18816, 16384, 4608;\n","recurrence_sentence":"The row polynomials satisfy (n^2-n-(2*n-2)*x)*P(n,x) = -(2*n^2-8*n+6-(10*n-14)*x+8*x^2)*P(n-1,x) + ((4*n-4)*x+(2*n-10)*x^2-4*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^2 - 8*n + 6 - (10*n - 14)*x + 8*x^2)*P[n - 1, x] + ((4*n - 4)*x + (2*n - 10)*x^2 - 4*x^3)*D[P[n - 1, x], x])/(n^2 - n - (2*n - 2)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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