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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":"be9e1e2ab0aabb01d3fe8481715ddca07db206cefa20102cca861c40c9cfd9db","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-35bd700768cac55f","role":"sequence","name":"Triangle read by rows: triangle of repeated stepped binomial coefficients.","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: 0, 2;\n  2: 0, 2, 3;\n  3: 0, 0, 3, 4;\n  4: 0, 0, 3, 8,  5;\n  5: 0, 0, 0, 4, 10,  6;\n  6: 0, 0, 0, 4, 15, 18,  7;\n  7: 0, 0, 0, 0,  5, 18, 21,  8;\n  8: 0, 0, 0, 0,  5, 24, 42, 32,  9;\n  9: 0, 0, 0, 0,  0,  6, 28, 48, 36, 10;\n","recurrence_sentence":"The row polynomials satisfy (n^2+3*n+2+(4*n^2+2*n)*x+(4*n^2-4*n)*x^2)*P(n,x) = -(n^3+2*n^2-n-2+(6*n^3+7*n^2-13*n-12)*x+(12*n^3-30*n+6)*x^2+(8*n^3-12*n^2-4*n+8)*x^3)*P(n-1,x) + ((2*n^2+6*n+4)*x+(10*n^2+18*n)*x^2+(16*n^2+12*n-8)*x^3+(8*n^2-4)*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[(-(n^3 + 2*n^2 - n - 2 + (6*n^3 + 7*n^2 - 13*n - 12)*x + (12*n^3 - 30*n + 6)*x^2 + (8*n^3 - 12*n^2 - 4*n + 8)*x^3)*P[n - 1, x] + ((2*n^2 + 6*n + 4)*x + (10*n^2 + 18*n)*x^2 + (16*n^2 + 12*n - 8)*x^3 + (8*n^2 - 4)*x^4)*D[P[n - 1, x], x])/(n^2 + 3*n + 2 + (4*n^2 + 2*n)*x + (4*n^2 - 4*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] = (-(n^3 + 2*n^2 - n - 2 + (6*n^3 + 7*n^2 - 13*n - 12)*x + (12*n^3 - 30*n + 6)*x^2 + (8*n^3 - 12*n^2 - 4*n + 8)*x^3)*v[i - 1] + ((2*n^2 + 6*n + 4)*x + (10*n^2 + 18*n)*x^2 + (16*n^2 + 12*n - 8)*x^3 + (8*n^2 - 4)*x^4)*deriv(v[i - 1], x))/(n^2 + 3*n + 2 + (4*n^2 + 2*n)*x + (4*n^2 - 4*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-35bd700768cac55f","steps":["\u03b8+1"],"edges":["e-1c588a07b00ef8dc"],"text":"A103633 \u2192 \u03b8+1"},"headline":"A103633 \u2192 \u03b8+1","chain":null}