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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":"4eb92129a7e9e6c5d8e639f0957b5012815decd99d77c86e15dcada8e0c47b1b","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-39736a4ef132a31d","role":"sequence","name":"basis_map_A013609 of Triangle of coefficients of Chebyshev's S(n,x+5) 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:       24,        8;\n  3:      212,      144,       24;\n  4:     1656,     1704,      576,       64;\n  5:    12102,    16704,     8544,     1920,      160;\n  6:    84816,   146928,   100800,    34240,     5760,     384;\n  7:   577576,  1204128,  1037040,   472320,   120000,   16128,     896;\n  8:  3851472,  9388144,  9733824,  5565120,  1895040,  384384,   43008,   2048;\n  9: 25275466, 70530624, 85530336, 58875264, 25163040, 6838272, 1154048, 110592, 4608;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (3*n+6+(n+2)*x)*P(n-1,x) + (8+6*x+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[((3*n + 6 + (n + 2)*x)*P[n - 1, x] + (8 + 6*x + x^2)*D[P[n - 1, x], x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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