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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":"d686b62d669641e630a8bd6a277db71b562266cde7f9a6a9773317545c99e54d","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:  0;\n   2:  1;\n   3: -2,   2;\n   4:  1,  -6,   3;\n   5:  2,   6, -12,   4;\n   6: -4,   4,  18, -20,    5;\n   7:  2, -18,   0,  40,  -30,    6;\n   8:  3,  18, -45, -20,   75,  -42,   7;\n   9: -6,   6,  72, -80,  -70,  126, -56,   8;\n  10:  3, -36, -18, 196, -105, -168, 196, -72, 9;\n","recurrence_sentence":"The row polynomials satisfy (2*n-4)*P(n,x) = -(n+1-(n+1)*x)*P(n-1,x) - (3+2*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[(-(n + 1 - (n + 1)*x)*P[n - 1, x] - (3 + 2*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)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = (-(n + 1 - (n + 1)*x)*v[i - 1] - (3 + 2*x - x^2)*deriv(v[i - 1], x))/(2*n - 4)); 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"],"edges":["e-7ddc15d309ad2a75","e-a1fc7f21c50978c7"],"text":"A053122 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A053122 \u2192 x \u21a6 x+1 \u2192 D","chain":null}