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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":"1c48f90e4938416455c48f814d8bd8e4707632e36476e97e8c89e3c89a5dd75b","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-63490e7c6342a0ed","role":"sequence","name":"veronese_section_2_0 of Triangle of coefficients in expansion of (1+3*x)^n.","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;\n  2: 1,  18;\n  3: 1,  54;\n  4: 1, 108,   243;\n  5: 1, 180,  1215;\n  6: 1, 270,  3645,   2916;\n  7: 1, 378,  8505,  20412;\n  8: 1, 504, 17010,  81648,  32805;\n  9: 1, 648, 30618, 244944, 295245;\n","recurrence_sentence":"The row polynomials satisfy -(2*n-2-(3*n^3-3*n)*x)*P(n,x) = -(2*n-2-(3*n^3-39*n+36)*x-(27*n^3+54*n^2-27*n-54)*x^2)*P(n-1,x) - (4*x-(6*n^2+18*n+48)*x^2+(54*n^2+162*n+108)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 2 - (3*n^3 - 39*n + 36)*x - (27*n^3 + 54*n^2 - 27*n - 54)*x^2)*P[n - 1, x] - (4*x - (6*n^2 + 18*n + 48)*x^2 + (54*n^2 + 162*n + 108)*x^3)*D[P[n - 1, x], x])/(-2*n + 2 + (3*n^3 - 3*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; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n - 2 - (3*n^3 - 39*n + 36)*x - (27*n^3 + 54*n^2 - 27*n - 54)*x^2)*v[i - 1] - (4*x - (6*n^2 + 18*n + 48)*x^2 + (54*n^2 + 162*n + 108)*x^3)*deriv(v[i - 1], x))/(-2*n + 2 + (3*n^3 - 3*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-6a5af5e1168bb679","steps":["Veronese 2","\u03b8+1"],"edges":["e-a3d90d8bb42bdd7e","e-f8b0bd1c63b20bae"],"text":"A013610 \u2192 Veronese 2 \u2192 \u03b8+1"},"headline":"A013610 \u2192 Veronese 2 \u2192 \u03b8+1","chain":null}