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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":"c13470a29900fe879329dfcf91507ba02a408e765f7ff3ff725e48fdf5da0d7f","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-51a2369b25b8e3ba","role":"sequence","name":"shift_x_plus_one of Triangle read by rows: T(n,k) = binomial(k,n-k).","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:  2,    2;\n  2:  3,    8,    3;\n  3:  5,   20,   18,    4;\n  4:  8,   44,   63,   32,     5;\n  5: 13,   90,  177,  144,    50,    6;\n  6: 21,  176,  441,  496,   275,   72,    7;\n  7: 34,  334, 1017, 1464,  1125,  468,   98,    8;\n  8: 55,  620, 2223, 3904,  3850, 2220,  735,  128,   9;\n  9: 89, 1132, 4671, 9688, 11685, 8658, 3969, 1088, 162, 10;\n","recurrence_sentence":"The row polynomials satisfy (10*n+10+(5*n^2+13*n+8)*x+(n^3+2*n^2+n)*x^2-(n^2+n)*x^3)*P(n,x) = -(20*n-60+(15*n^2-39*n-68)*x+(2*n^3-4*n^2-36*n-22)*x^2-(5*n^2+3*n-4)*x^3+(2*n+2)*x^4)*P(n-1,x) + (25+(25*n+45)*x+(5*n^2+35*n+18)*x^2+(6*n^2+6*n-8)*x^3+(n^2-5*n-7)*x^4-(n+1)*x^5)*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[(-(20*n - 60 + (15*n^2 - 39*n - 68)*x + (2*n^3 - 4*n^2 - 36*n - 22)*x^2 - (5*n^2 + 3*n - 4)*x^3 + (2*n + 2)*x^4)*P[n - 1, x] + (25 + (25*n + 45)*x + (5*n^2 + 35*n + 18)*x^2 + (6*n^2 + 6*n - 8)*x^3 + (n^2 - 5*n - 7)*x^4 - (n + 1)*x^5)*D[P[n - 1, x], x])/(10*n + 10 + (5*n^2 + 13*n + 8)*x + (n^3 + 2*n^2 + n)*x^2 - (n^2 + n)*x^3)]];\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] = (-(20*n - 60 + (15*n^2 - 39*n - 68)*x + (2*n^3 - 4*n^2 - 36*n - 22)*x^2 - (5*n^2 + 3*n - 4)*x^3 + (2*n + 2)*x^4)*v[i - 1] + (25 + (25*n + 45)*x + (5*n^2 + 35*n + 18)*x^2 + (6*n^2 + 6*n - 8)*x^3 + (n^2 - 5*n - 7)*x^4 - (n + 1)*x^5)*deriv(v[i - 1], x))/(10*n + 10 + (5*n^2 + 13*n + 8)*x + (n^3 + 2*n^2 + n)*x^2 - (n^2 + n)*x^3)); 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-51a2369b25b8e3ba","steps":["\u03b8+1"],"edges":["e-a50e6dda87386652"],"text":"A154929 \u2192 \u03b8+1"},"headline":"A154929 \u2192 \u03b8+1","chain":null}