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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":"5f0f8be8a8f5caf566ab61186a3253fafb7345f9cd3ebb498bd28c02a6ed2c6d","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-d2e17233522b2755","role":"sequence","name":"Triangle of 2-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:    2;\n  2:    7,      8;\n  3:   18,     66,      24;\n  4:   41,    342,     393,       64;\n  5:   88,   1436,    3624,     1892,      160;\n  6:  183,   5364,   25266,    28788,     8055,      384;\n  7:  374,  18654,  149340,   312380,   192270,    31686,     896;\n  8:  757,  61946,  793227,  2756620,  3105995,  1144698,  118069,   2048;\n  9: 1524, 199464, 3916260, 21186552, 39310620, 26648856, 6286476, 423240, 4608;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = -(2*n+2+(2*n^2-2)*x+(n^3-n)*x^2)*P(n-2,x) - ((n+1)*x+(n^2-1)*x^2-(n^2+n)*x^3)*d/dx P(n-2,x) + (3*n+1+2*n^2*x)*P(n-1,x) + (n*x-n*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n + 2 + (2*n^2 - 2)*x + (n^3 - n)*x^2)*P[n - 2, x] - ((n + 1)*x + (n^2 - 1)*x^2 - (n^2 + n)*x^3)*D[P[n - 2, x], x] + (3*n + 1 + 2*n^2*x)*P[n - 1, x] + (n*x - n*x^2)*D[P[n - 1, x], x])/(n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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