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Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"8609d50ba9ad45b7433693c2d5bacafa7ed0584e36bf44d5a56785816fdd7e92","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-5fc5274bb6446a33","role":"sequence","name":"shift_x_plus_one of chebyshev_t","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1,   2;\n  2: 1,   8,    6;\n  3: 1,  18,   36,    16;\n  4: 1,  32,  120,   128,    40;\n  5: 1,  50,  300,   560,   400,     96;\n  6: 1,  72,  630,  1792,  2160,   1152,    224;\n  7: 1,  98, 1176,  4704,  8400,   7392,   3136,   512;\n  8: 1, 128, 2016, 10752, 26400,  33792,  23296,  8192,  1152;\n  9: 1, 162, 3240, 22176, 71280, 123552, 122304, 69120, 20736, 2560;\n","recurrence_sentence":"The row polynomials satisfy -(3*n-3-(n^3-3*n^2+2*n)*x)*P(n,x) = -(3*n-3-(n^3-3*n^2-4*n+6)*x-(n^3-2*n^2-n+2)*x^2)*P(n-1,x) - (6*x-(2*n^2-2*n-7)*x^2-(n^2-n-2)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(3*n - 3 - (n^3 - 3*n^2 - 4*n + 6)*x - (n^3 - 2*n^2 - n + 2)*x^2)*P[n - 1, x] - (6*x - (2*n^2 - 2*n - 7)*x^2 - (n^2 - n - 2)*x^3)*D[P[n - 1, x], x])/(-3*n + 3 + (n^3 - 3*n^2 + 2*n)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(3*n - 3 - (n^3 - 3*n^2 - 4*n + 6)*x - (n^3 - 2*n^2 - n + 2)*x^2)*v[i - 1] - (6*x - (2*n^2 - 2*n - 7)*x^2 - (n^2 - n - 2)*x^3)*deriv(v[i - 1], x))/(-3*n + 3 + (n^3 - 3*n^2 + 2*n)*x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-9ae8d431f7b33f22","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-298ee5f2391f30e9","e-ea7686e5aa56d7b8"],"text":"chebyshev_t \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"chebyshev_t \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}