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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":"632e16fc8d442ca4711d825350c37a7fdfd494e78ff9316e175d8b2bcd65c090","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-35f03041aed96954","role":"sequence","name":"Triangle of 5-Eulerian numbers.","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:    5;\n  2:   16,     50;\n  3:   39,    342,      375;\n  4:   86,   1572,     4578,      2500;\n  5:  181,   6092,    34818,     49124,      15625;\n  6:  372,  21534,   212376,    570588,     462180,      93750;\n  7:  755,  72042,  1140213,   5118284,    7765725,    3983946,     546875;\n  8: 1522, 232736,  5641482,  39273976,   97777190,   93119712,   32262622,   3125000;\n  9: 3057, 734736, 26406072, 271589640, 1028495370, 1590499008, 1018578624, 249354408, 17578125;\n","recurrence_sentence":"The row polynomials satisfy (n+3)*P(n,x) = -(2*n+8+(2*n^2+12*n+16)*x+(n^3+9*n^2+26*n+24)*x^2)*P(n-2,x) - ((n+4)*x+(n^2+6*n+8)*x^2-(n^2+7*n+12)*x^3)*d/dx P(n-2,x) + (3*n+10+(2*n^2+12*n+18)*x)*P(n-1,x) + ((n+3)*x-(n+3)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 5.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 5;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n + 8 + (2*n^2 + 12*n + 16)*x + (n^3 + 9*n^2 + 26*n + 24)*x^2)*P[n - 2, x] - ((n + 4)*x + (n^2 + 6*n + 8)*x^2 - (n^2 + 7*n + 12)*x^3)*D[P[n - 2, x], x] + (3*n + 10 + (2*n^2 + 12*n + 18)*x)*P[n - 1, x] + ((n + 3)*x - (n + 3)*x^2)*D[P[n - 1, x], x])/(n + 3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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