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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":"b9c1b23c07f07b4d4f24c68fa74cfd8b3729b22ffc05d9cfba8a2cb6716d24e9","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-f7e7f08472ec987a","role":"sequence","name":"Triangle of coefficients of Chebyshev's S(n,x-2) = U(n,x/2-1) polynomials (exponents of x in increasing order).","uses":{"claim":"real_rooted","from_index":1,"region":{}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)"}]}],"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:   -4,     2;\n   4:   10,   -12,     3;\n   5:  -20,    42,   -24,     4;\n   6:   35,  -112,   108,   -40,     5;\n   7:  -56,   252,  -360,   220,   -60,     6;\n   8:   84,  -504,   990,  -880,   390,   -84,    7;\n   9: -120,   924, -2376,  2860, -1820,   630, -112,    8;\n  10:  165, -1584,  5148, -8008,  6825, -3360,  952, -144, 9;\n","recurrence_sentence":"The row polynomials satisfy (2*n-4)*P(n,x) = -(2*n+2-(n+1)*x)*P(n-1,x) - (4*x-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[(-(2*n + 2 - (n + 1)*x)*P[n - 1, x] - (4*x - x^2)*D[P[n - 1, x], x])/(2*n - 4)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); 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