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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":"c09415281f8b116713a7da85210d92d5f2bbdacd890391a7de96f0456aa87379","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-2feb6dc01f424b0a","role":"sequence","name":"Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....","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:  1;\n  1:  3,    2;\n  2:  6,   17,     6;\n  3: 10,   61,    98,     24;\n  4: 15,  156,   533,    634,     120;\n  5: 21,  330,  1895,   4756,    4644,      720;\n  6: 28,  617,  5290,  22081,   45244,    38268,     5040;\n  7: 36, 1057, 12586,  77721,  262724,   464444,   351504,    40320;\n  8: 45, 1696, 26698, 227220, 1128029,  3268700,  5154876,  3566736,   362880;\n  9: 55, 2586, 51918, 580440, 3946719, 16695750, 42931916, 61752456, 39659040, 3628800;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = -(n+1+(2*n^2+2*n)*x+(n^3+n^2)*x^2)*P(n-2,x) + (2*n+1+(2*n^2+2*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n + 1 + (2*n^2 + 2*n)*x + (n^3 + n^2)*x^2)*P[n - 2, x] + (2*n + 1 + (2*n^2 + 2*n)*x)*P[n - 1, x])/(n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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