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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":"de60b31b3759e16a9b08e0fecb5a2a4eb9f35a359b82abe8d88664f2781a3d88","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-2de752bc9bc14cd4","role":"sequence","name":"shift_x_plus_one of 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":{"lower":{"closed":false,"value":"-1"},"upper":{"closed":false,"value":"3"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-1, 3)"}]}],"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:  1;\n   2:  0,   1;\n   3: -2,   0,   1;\n   4:  2,  -5,   0,   1;\n   5:  1,   8,  -9,   0,   1;\n   6: -4,   0,  20, -14,   0,    1;\n   7:  3, -16,  -9,  40, -20,    0,   1;\n   8:  2,  21, -36, -35,  70,  -27,   0,   1;\n   9: -6,   0,  75, -56, -90,  112, -35,   0, 1;\n  10:  4, -33, -36, 190, -56, -189, 168, -44, 0, 1;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3-6*n^2+4*n+(10*n-10)*x-(2*n-2)*x^2)*P(n,x) = -(n^3-4*n-(n^3-4*n)*x-5*n*x^2+n*x^3)*P(n-1,x) - (3*n^2-3*n+6+(2*n^2-2*n+13)*x-(n^2-n-1)*x^2-5*x^3+x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 4*n - (n^3 - 4*n)*x - 5*n*x^2 + n*x^3)*P[n - 1, x] - (3*n^2 - 3*n + 6 + (2*n^2 - 2*n + 13)*x - (n^2 - n - 1)*x^2 - 5*x^3 + x^4)*D[P[n - 1, x], x])/(2*n^3 - 6*n^2 + 4*n + (10*n - 10)*x - (2*n - 2)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(n^3 - 4*n - (n^3 - 4*n)*x - 5*n*x^2 + n*x^3)*v[i - 1] - (3*n^2 - 3*n + 6 + (2*n^2 - 2*n + 13)*x - (n^2 - n - 1)*x^2 - 5*x^3 + x^4)*deriv(v[i - 1], x))/(2*n^3 - 6*n^2 + 4*n + (10*n - 10)*x - (2*n - 2)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-f7e7f08472ec987a","steps":["x \u21a6 x+1","D+1"],"edges":["e-7ddc15d309ad2a75","e-ba9336e6375434bf"],"text":"A053122 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A053122 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}