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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":"b32a70de897dab86f22c61a4b53ee9f0359fd6f5d6d92e2f44bf98b2afa9442c","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-3a7609bd7546c802","role":"sequence","name":"Triangle of coefficients of Chebyshev's U(n,2*x-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:    4;\n   3:  -16,     32;\n   4:   40,   -192,     192;\n   5:  -80,    672,   -1536,     1024;\n   6:  140,  -1792,    6912,   -10240,     5120;\n   7: -224,   4032,  -23040,    56320,   -61440,     24576;\n   8:  336,  -8064,   63360,  -225280,   399360,   -344064,   114688;\n   9: -480,  14784, -152064,   732160, -1863680,   2580480, -1835008,   524288;\n  10:  660, -25344,  329472, -2050048,  6988800, -13762560, 15597568, -9437184, 2359296;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = -(n+1-(2*n+2)*x)*P(n-1,x) - (2*x-2*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 4.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n + 1 - (2*n + 2)*x)*P[n - 1, x] - (2*x - 2*x^2)*D[P[n - 1, x], x])/(n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 4; for(n = 3, N, my(i = n); v[i] = (-(n + 1 - (2*n + 2)*x)*v[i - 1] - (2*x - 2*x^2)*deriv(v[i - 1], x))/(n - 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-3a7609bd7546c802","steps":["D"],"edges":["e-38acfb7dea411f23"],"text":"A053124 \u2192 D"},"headline":"A053124 \u2192 D","chain":null}