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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":"dcec462d9416052538eae37ba0bd6f9089d4b6e9cde8a88b40cbc762ab940269","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-9496336e2b29e28e","role":"sequence","name":"Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).","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,  6;\n  2: 1, 10,  15;\n  3: 1, 14,  42,   28;\n  4: 1, 18,  81,  120,   45;\n  5: 1, 22, 132,  308,  275,    66;\n  6: 1, 26, 195,  624,  910,   546,    91;\n  7: 1, 30, 270, 1100, 2250,  2268,   980,   120;\n  8: 1, 34, 357, 1768, 4675,  6732,  4998,  1632,  153;\n  9: 1, 38, 456, 2660, 8645, 16302, 17556, 10032, 2565, 190;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2-2*n+(8*n^3-2*n)*x)*P(n,x) = (4*n^2-2*n+(8*n^3+24*n^2-14*n)*x+(24*n^3+32*n^2-18*n-2)*x^2)*P(n-1,x) - (2*n*x+(4*n^2+16*n+1)*x^2+(16*n^2+32*n+4)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 2*n + (8*n^3 + 24*n^2 - 14*n)*x + (24*n^3 + 32*n^2 - 18*n - 2)*x^2)*P[n - 1, x] - (2*n*x + (4*n^2 + 16*n + 1)*x^2 + (16*n^2 + 32*n + 4)*x^3)*D[P[n - 1, x], x])/(4*n^2 - 2*n + (8*n^3 - 2*n)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^2 - 2*n + (8*n^3 + 24*n^2 - 14*n)*x + (24*n^3 + 32*n^2 - 18*n - 2)*x^2)*v[i - 1] - (2*n*x + (4*n^2 + 16*n + 1)*x^2 + (16*n^2 + 32*n + 4)*x^3)*deriv(v[i - 1], x))/(4*n^2 - 2*n + (8*n^3 - 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-9496336e2b29e28e","steps":["\u03b8+1"],"edges":["e-d6813283d076dba7"],"text":"A082985 \u2192 \u03b8+1"},"headline":"A082985 \u2192 \u03b8+1","chain":null}