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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":"2a403741eacd861c3d7c997c7b83f373cdb558124b5c25cfef08536bf0f3242d","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-0135f5d413e6cc58","role":"sequence","name":"basis_map_A007318 of 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:    4,     6;\n  2:   11,    30,     15;\n  3:   29,   112,    105,     28;\n  4:   76,   378,    513,    264,     45;\n  5:  199,  1210,   2145,   1628,    550,     66;\n  6:  521,  3744,   8190,   8216,   4160,   1014,     91;\n  7: 1364, 11310,  29430,  36720,  24825,   9198,   1715,   120;\n  8: 3571, 33558, 101235, 151096, 127925,  63852,  18326,  2720,  153;\n  9: 9349, 98192, 336927, 585428, 595840, 374718, 146034, 33744, 4104, 190;\n","recurrence_sentence":"The row polynomials satisfy (20*n-10+(20*n^2+2*n-6)*x+(8*n^3-2*n)*x^2)*P(n,x) = (80*n-40+(100*n^2+98*n-74)*x+(32*n^3+120*n^2+4*n-36)*x^2+(24*n^3+32*n^2-18*n-2)*x^3)*P(n-1,x) - (25+(50*n+75)*x+(20*n^2+130*n+79)*x^2+(36*n^2+112*n+33)*x^3+(16*n^2+32*n+4)*x^4)*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[((80*n - 40 + (100*n^2 + 98*n - 74)*x + (32*n^3 + 120*n^2 + 4*n - 36)*x^2 + (24*n^3 + 32*n^2 - 18*n - 2)*x^3)*P[n - 1, x] - (25 + (50*n + 75)*x + (20*n^2 + 130*n + 79)*x^2 + (36*n^2 + 112*n + 33)*x^3 + (16*n^2 + 32*n + 4)*x^4)*D[P[n - 1, x], x])/(20*n - 10 + (20*n^2 + 2*n - 6)*x + (8*n^3 - 2*n)*x^2)]];\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] = ((80*n - 40 + (100*n^2 + 98*n - 74)*x + (32*n^3 + 120*n^2 + 4*n - 36)*x^2 + (24*n^3 + 32*n^2 - 18*n - 2)*x^3)*v[i - 1] - (25 + (50*n + 75)*x + (20*n^2 + 130*n + 79)*x^2 + (36*n^2 + 112*n + 33)*x^3 + (16*n^2 + 32*n + 4)*x^4)*deriv(v[i - 1], x))/(20*n - 10 + (20*n^2 + 2*n - 6)*x + (8*n^3 - 2*n)*x^2)); 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":["x \u21a6 x+1","\u03b8+1"],"edges":["e-557cf165357d4e1f","e-22e4e8d85f253659"],"text":"A082985 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A082985 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}