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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":"86ec78cf83ba786581cfc4fba9c3cf171776add9fef689d3e191245aa3f8dce7","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-15df20466f2d1f05","role":"sequence","name":"Stirling-like number triangle defined by sequence A000217.","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,       3;\n  3: 1,    26,      30,        4;\n  4: 1,    80,     219,       80,        5;\n  5: 1,   242,    1434,     1092,      175,       6;\n  6: 1,   728,    8967,    12832,     3990,     336,       7;\n  7: 1,  2186,   54894,   140276,    75890,   11844,     588,     8;\n  8: 1,  6560,  332643,  1475952,  1313695,  339792,   30282,   960,    9;\n  9: 1, 19682, 2005698, 15203044, 21550365, 8712066, 1244320, 69168, 1485, 10;\n","recurrence_sentence":"The row polynomials satisfy 2*P(n,x) = -2*x^2*P(n-2,x) - 3*x^2*d/dx P(n-2,x) - x^3*d^2/dx^2 P(n-2,x) + (2+4*x)*P(n-1,x) + 4*x*d/dx P(n-1,x) + x^2*d^2/dx^2 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[(-2*x^2*P[n - 2, x] - 3*x^2*D[P[n - 2, x], x] - x^3*D[P[n - 2, x], {x, 2}] + (2 + 4*x)*P[n - 1, x] + 4*x*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}])/(2)]];\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] = (-2*x^2*v[i - 2] - 3*x^2*deriv(v[i - 2], x) - x^3*deriv(deriv(v[i - 2], x), x) + (2 + 4*x)*v[i - 1] + 4*x*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), 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-15df20466f2d1f05","steps":["\u03b8+1"],"edges":["e-b5c2a9462edbac22"],"text":"A080248 \u2192 \u03b8+1"},"headline":"A080248 \u2192 \u03b8+1","chain":null}