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for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"999c66a94187f7a7b851238ef1ebedeb6a6f7ba40fb0733dd41aa3af36ba7027","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-0b6e037cc63716ca","role":"sequence","name":"Triangle of coefficients of shifted Chebyshev's S(n,x-2) = U(n,x/2-1) polynomials (exponents of x in decreasing 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:  1;\n   2: -1,  -2;\n   3:  0,   2,   3;\n   4:  1,   2,  -2,   -4;\n   5: -1,  -6,  -9,    0,    5;\n   6:  0,   4,  18,   24,    5,   -6;\n   7:  1,   4,  -4,  -36,  -49,  -14,    7;\n   8: -1, -10, -30,  -20,   50,   84,   28,  -8;\n   9:  0,   6,  45,  120,  115,  -36, -126, -48,  9;\n  10:  1,   6,  -6, -120, -335, -358,  -48, 168, 75, -10;\n","recurrence_sentence":"The row polynomials satisfy (2*n-2)*P(n,x) = -(4*n-6+(6*n-8)*x)*P(n-1,x) + (3+7*x+4*x^2)*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[(-(4*n - 6 + (6*n - 8)*x)*P[n - 1, x] + (3 + 7*x + 4*x^2)*D[P[n - 1, x], x])/(2*n - 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] = (-(4*n - 6 + (6*n - 8)*x)*v[i - 1] + (3 + 7*x + 4*x^2)*deriv(v[i - 1], x))/(2*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-0b6e037cc63716ca","steps":["x \u21a6 x+1"],"edges":["e-fb33150e82d07f76"],"text":"A053123 \u2192 x \u21a6 x+1"},"headline":"A053123 \u2192 x \u21a6 x+1","chain":null}