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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":"53209827c373dafdd1f4158026f910107af09719e40d0374aba4b61246724c1d","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-181be2f2eb86cd8e","role":"sequence","name":"Stirling2 triangle with scaled diagonals (powers of 3).","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:       1;\n  2:       9,        2;\n  3:      63,       36,        3;\n  4:     405,      450,       90,        4;\n  5:    2511,     4860,     1755,      180,       5;\n  6:   15309,    48762,    28350,     5040,     315,      6;\n  7:   92583,   469476,   413343,   113400,   11970,    504,     7;\n  8:  557685,  4410450,  5664330,  2252124,  357210,  24948,   756,    8;\n  9: 3352671, 40809420, 74587635, 41334300, 9244935, 952560, 47250, 1080, 9;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(18+6*x+x^2)*P(n-2,x) - (9*x+3*x^2)*d/dx P(n-2,x) + (9+2*x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[-(18 + 6*x + x^2)*P[n - 2, x] - (9*x + 3*x^2)*D[P[n - 2, x], x] + (9 + 2*x)*P[n - 1, x] + 3*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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