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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":"c7a60003b73406e203418d68623b4db594b43867c62ef0181c329093efe816a4","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-86fb6463ff8368f2","role":"sequence","name":"Triangle of third-order Eulerian numbers: 3-Stirling permutations enumerated by ascents.","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:    3;\n  2:   12,      30;\n  3:   33,     282,      315;\n  4:   78,    1572,     5490,       3780;\n  5:  171,    6900,    52614,     106740,      51975;\n  6:  360,   26514,   376056,    1585980,    2179800,      810810;\n  7:  741,   93942,  2259873,   16955244,   46780875,    47527830,    14189175;\n  8: 1506,  315912, 12139578,  148052664,  704566350,  1399298760,  1111597830,   275675400;\n  9: 3039, 1026048, 60405012, 1128729816, 8483912190, 28395642960, 43175602260, 27889029000, 5892561675;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = -(2*n+(6*n^2-8*n)*x+(9*n^3-21*n^2+12*n)*x^2)*P(n-2,x) - (n*x+(3*n^2-4*n)*x^2-(3*n^2-3*n)*x^3)*d/dx P(n-2,x) + (3*n-2+(6*n^2-8*n+2)*x)*P(n-1,x) + ((n-1)*x-(n-1)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n + (6*n^2 - 8*n)*x + (9*n^3 - 21*n^2 + 12*n)*x^2)*P[n - 2, x] - (n*x + (3*n^2 - 4*n)*x^2 - (3*n^2 - 3*n)*x^3)*D[P[n - 2, x], x] + (3*n - 2 + (6*n^2 - 8*n + 2)*x)*P[n - 1, x] + ((n - 1)*x - (n - 1)*x^2)*D[P[n - 1, x], x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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