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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":"0944d566da0feb394bf241ed3748478830db98c7c47ccd96b57e2fc03282e7cb","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-4f1c1e4637c4f22d","role":"sequence","name":"basis_map_A013609 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:       2;\n  2:      20,        8;\n  3:     146,      120,       24;\n  4:     940,     1176,      480,       64;\n  5:    5656,     9520,     5904,     1600,      160;\n  6:   32620,    69048,    57600,    23680,     4800,     384;\n  7:  182742,   466200,   489240,   270400,    83040,   13440,    896;\n  8: 1002300,  2992920,  3787200,  2632960,  1086400,  266112,  35840,  2048;\n  9: 5409510, 18507600, 27425520, 22993600, 11931360, 3924480, 799232, 92160, 4608;\n","recurrence_sentence":"The row polynomials satisfy (4*n-4)*P(n,x) = (10*n+20+(4*n+8)*x)*P(n-1,x) + (21+20*x+4*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((10*n + 20 + (4*n + 8)*x)*P[n - 1, x] + (21 + 20*x + 4*x^2)*D[P[n - 1, x], x])/(4*n - 4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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