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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":"0a537839f25dcae6b1ffa0b7ffde0468bdb742c107c4c6165b5a7d38e258e874","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-06404fc73ac5a6a1","role":"sequence","name":"reversal of Triangle read by rows: 3-Stirling numbers of the second kind.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:  0;\n   2:  3;\n   3:  7,   18;\n   4: 12,   74,    81;\n   5: 18,  194,   525,    324;\n   6: 25,  410,  1980,   3124,    1215;\n   7: 33,  760,  5670,  16324,   16835,    4374;\n   8: 42, 1288, 13650,  61684,  118860,   85182,   15309;\n   9: 52, 2044, 29106, 189084,  581490,  798342,  412825,   52488;\n  10: 63, 3084, 56700, 499548, 2235975, 4985070, 5069820, 1939688, 177147;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = -(n+1+(2*n^2-2)*x+(n^3-n)*x^2)*P(n-2,x) + ((n+1)*x^2+(n^2+n)*x^3)*d/dx P(n-2,x) + (2*n+1+2*n^2*x)*P(n-1,x) - n*x^2*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 3.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n + 1 + (2*n^2 - 2)*x + (n^3 - n)*x^2)*P[n - 2, x] + ((n + 1)*x^2 + (n^2 + n)*x^3)*D[P[n - 2, x], x] + (2*n + 1 + 2*n^2*x)*P[n - 1, x] - n*x^2*D[P[n - 1, x], x])/(n)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); 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