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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":"dfe2261f7442eb3a1896f28eb7b2f4015bc1220e18dbb79267e125c4b760aaad","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-6ad01dc6e705cdb6","role":"sequence","name":"Pascal-(1,5,1) array.","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:  7,    2;\n  3: 13,   26,     3;\n  4: 19,  122,    57,     4;\n  5: 25,  290,   435,   100,     5;\n  6: 31,  530,  1785,  1060,   155,     6;\n  7: 37,  842,  4755,  6340,  2105,   222,    7;\n  8: 43, 1226,  9993, 24580, 16655,  3678,  301,   8;\n  9: 49, 1682, 18147, 69604, 87005, 36294, 5887, 392, 9;\n","recurrence_sentence":"The row polynomials satisfy -(6*n^2-17*n+11-(6*n^2-4*n-2)*x+(n-1)*x^2)*P(n,x) = -(6*n^2-11*n+5+(66*n^2-260*n+160)*x-(29*n-37)*x^2+2*x^3)*P(n-1,x) + ((6*n-5)*x+(132*n-111)*x^2+(6*n-27)*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[(-(6*n^2 - 11*n + 5 + (66*n^2 - 260*n + 160)*x - (29*n - 37)*x^2 + 2*x^3)*P[n - 1, x] + ((6*n - 5)*x + (132*n - 111)*x^2 + (6*n - 27)*x^3 - x^4)*D[P[n - 1, x], x])/(-6*n^2 + 17*n - 11 + (6*n^2 - 4*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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