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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":"125d0cf20b11d94e6d1565658c81cde2f0d5643f67e5459fdc60e55ecfb56070","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-f25cd955a876cde3","role":"sequence","name":"Pascal-(1,2,1) array read by antidiagonals.","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:  1;\n  2:  4,   2;\n  3:  7,  14,    3;\n  4: 10,  44,   30,    4;\n  5: 13,  92,  138,   52,     5;\n  6: 16, 158,  408,  316,    80,    6;\n  7: 19, 242,  921, 1228,   605,  114,    7;\n  8: 22, 344, 1758, 3544,  2930, 1032,  154,   8;\n  9: 25, 464, 3000, 8344, 10430, 6000, 1624, 200, 9;\n","recurrence_sentence":"The row polynomials satisfy -(3*n^2-8*n+5-(3*n^2-n-2)*x+(n-1)*x^2)*P(n,x) = -(3*n^2-5*n+2+(15*n^2-59*n+28)*x-(14*n-16)*x^2+2*x^3)*P(n-1,x) + ((3*n-2)*x+(30*n-21)*x^2+(3*n-12)*x^3-x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(3*n^2 - 5*n + 2 + (15*n^2 - 59*n + 28)*x - (14*n - 16)*x^2 + 2*x^3)*P[n - 1, x] + ((3*n - 2)*x + (30*n - 21)*x^2 + (3*n - 12)*x^3 - x^4)*D[P[n - 1, x], x])/(-3*n^2 + 8*n - 5 + (3*n^2 - n - 2)*x - (n - 1)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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