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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":"1bb25c71af707f3eb2194fa068753eb28eb2718d8d34bf45710351b4bad1ba6a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-f00f6de2d4bbc6f5","role":"sequence","name":"Triangle read by rows: the reversed x = 1+q Narayana triangle at m=2.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:       0;\n   2:       1;\n   3:       8,        2;\n   4:      55,       30,        3;\n   5:     364,      312,       72,       4;\n   6:    2380,     2800,     1050,     140,       5;\n   7:   15504,    23256,    12240,    2720,     240,      6;\n   8:  100947,   184338,   125685,   39900,    5985,    378,     7;\n   9:  657800,  1416800,  1190112,  495880,  107800,  11760,   560,   8;\n  10: 4292145, 10656360, 10656360, 5525520, 1593900, 255024, 21252, 792, 9;\n","recurrence_sentence":"The row polynomials satisfy (4*n^3-6*n^2-4*n)*P(n,x) = (27*n^3-54*n^2+33*n-6+(12*n^3-32*n^2+16*n)*x)*P(n-1,x) - ((9*n^2-9*n+2)*x+(8*n^2-6*n)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((27*n^3 - 54*n^2 + 33*n - 6 + (12*n^3 - 32*n^2 + 16*n)*x)*P[n - 1, x] - ((9*n^2 - 9*n + 2)*x + (8*n^2 - 6*n)*x^2)*D[P[n - 1, x], x])/(4*n^3 - 6*n^2 - 4*n)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); 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