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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":"5552bcd03f97d706d054fc85dc7f9a32928c95921ac63597b19b9fd7d45100a3","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-39578f69b35e8349","role":"sequence","name":"basis_map_A007318 of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.","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:     6,      2;\n  1:    21,     16,      3;\n  2:    60,     74,     30,      4;\n  3:   156,    268,    171,     48,      5;\n  4:   384,    848,    744,    324,     70,      6;\n  5:   912,   2464,   2760,   1640,    545,     96,     7;\n  6:  2112,   6752,   9216,   6960,   3140,    846,   126,     8;\n  7:  4800,  17728,  28560,  26208,  14980,   5460,  1239,   160,    9;\n  8: 10752,  45056,  83712,  90496,  62720,  28896,  8848,  1736,  198,  10;\n  9: 23808, 111616, 235008, 292608, 238560, 133056, 51408, 13584, 2349, 240, 11;\n","recurrence_sentence":"The row polynomials satisfy (3*n+1+(n+1)*x)*P(n,x) = (6*n+8+(5*n+8)*x+(n+2)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 6 + 2*x.","mathematica":"P[0, x_] := 6 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((6*n + 8 + (5*n + 8)*x + (n + 2)*x^2)*P[n - 1, x])/(3*n + 1 + (n + 1)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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