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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":"f9b71a4113becdb1dbfe2b50ae4ac8eabc224ae1223e563b6434b10a0979811a","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-0778fc04c5956f01","role":"sequence","name":"basis_map_A013610 of Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).","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:      30,       18;\n  3:     219,      270,       81;\n  4:    1410,     2646,     1620,       324;\n  5:    8484,    21420,    19926,      8100,     1215;\n  6:   48930,   155358,   194400,    119880,    36450,     4374;\n  7:  274113,  1048950,  1651185,   1368900,   630585,   153090,    15309;\n  8: 1503450,  6734070, 12781800,  13329360,  8249850,  3031182,   612360,   52488;\n  9: 8114265, 41642100, 92561130, 116405100, 90603765, 44702280, 13655628, 2361960, 177147;\n","recurrence_sentence":"The row polynomials satisfy (2*n-2)*P(n,x) = (5*n+10+(3*n+6)*x)*P(n-1,x) + (7+10*x+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[((5*n + 10 + (3*n + 6)*x)*P[n - 1, x] + (7 + 10*x + 3*x^2)*D[P[n - 1, x], x])/(2*n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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