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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":"ea2ed7811a78e7b5996c1a9ce57a60e198ade05d58367a7c5d442377a32e165b","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-117b55f7953e214b","role":"sequence","name":"basis_map_A013610 of Square the entries of Pascal's triangle.","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:     18,      18;\n  3:     90,     216,       81;\n  4:    420,    1620,     1620,      324;\n  5:   1890,   10080,    17010,     9720,      1215;\n  6:   8316,   56700,   136080,   136080,     51030,     4374;\n  7:  36036,  299376,   935550,  1360800,    918540,   244944,    15309;\n  8: 154440, 1513512,  5837832, 11226600,  11226600,  5511240,  1102248,   52488;\n  9: 656370, 7413120, 34054020, 81729648, 109459350, 80831520, 30311820, 4723920, 177147;\n","recurrence_sentence":"The row polynomials satisfy (n^2-2*n+1)*P(n,x) = (4*n^2-6*n+2+(9*n^2-18*n+6)*x)*P(n-1,x) - ((2*n-1)*x+(6*n-3)*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[((4*n^2 - 6*n + 2 + (9*n^2 - 18*n + 6)*x)*P[n - 1, x] - ((2*n - 1)*x + (6*n - 3)*x^2)*D[P[n - 1, x], x])/(n^2 - 2*n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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