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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":"93c613f9249be72e0746623b58267d94b1cd4b666264372702c52e21b6b21cbe","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-4c402d5a817396fa","role":"sequence","name":"Triangle of 3-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:    3;\n  2:   10,     18;\n  3:   25,    134,      81;\n  4:   56,    652,    1128,      324;\n  5:  119,   2628,    9402,     7636,      1215;\n  6:  246,   9550,   61260,   100860,     45470,      4374;\n  7:  501,  32586,  345315,   994380,    903195,    248874,    15309;\n  8: 1012, 106776, 1772292,  8194840,  12877020,   7161288,  1283884,   52488;\n  9: 2035, 340568, 8530644, 59878648, 148966970, 142843560, 52045252, 6341576, 177147;\n","recurrence_sentence":"The row polynomials satisfy (n+1)*P(n,x) = -(2*n+4+(2*n^2+4*n)*x+(n^3+3*n^2+2*n)*x^2)*P(n-2,x) - ((n+2)*x+(n^2+2*n)*x^2-(n^2+3*n+2)*x^3)*d/dx P(n-2,x) + (3*n+4+(2*n^2+4*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 + 4 + (2*n^2 + 4*n)*x + (n^3 + 3*n^2 + 2*n)*x^2)*P[n - 2, x] - ((n + 2)*x + (n^2 + 2*n)*x^2 - (n^2 + 3*n + 2)*x^3)*D[P[n - 2, x], x] + (3*n + 4 + (2*n^2 + 4*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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