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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":"674689d55e56dee56afb391307c2157b7f0933cca0f5099079bedae5eeaadaf7","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-de4284a2d5d7d0a6","role":"sequence","name":"A Motzkin related triangle.","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, 9,  4;\n  4: 0, 0, 6, 24,   5;\n  5: 0, 0, 0, 40,  50,   6;\n  6: 0, 0, 0, 20, 150,  90,    7;\n  7: 0, 0, 0,  0, 175, 420,  147,    8;\n  8: 0, 0, 0,  0,  70, 840,  980,  224,   9;\n  9: 0, 0, 0,  0,   0, 756, 2940, 2016, 324, 10;\n","recurrence_sentence":"The row polynomials satisfy -(n^2+3*n+2-(n^2+n)*x)*P(n,x) = (4*n^2-4-(7*n^2+5*n)*x+(3*n^2+n)*x^2)*P(n-1,x) - ((8*n+8)*x-(10*n+6)*x^2+(2*n+1)*x^3)*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[((4*n^2 - 4 - (7*n^2 + 5*n)*x + (3*n^2 + n)*x^2)*P[n - 1, x] - ((8*n + 8)*x - (10*n + 6)*x^2 + (2*n + 1)*x^3)*D[P[n - 1, x], x])/(-n^2 - 3*n - 2 + (n^2 + n)*x)]];\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] = ((4*n^2 - 4 - (7*n^2 + 5*n)*x + (3*n^2 + n)*x^2)*v[i - 1] - ((8*n + 8)*x - (10*n + 6)*x^2 + (2*n + 1)*x^3)*deriv(v[i - 1], x))/(-n^2 - 3*n - 2 + (n^2 + 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-de4284a2d5d7d0a6","steps":["\u03b8+1"],"edges":["e-8366440628a6d603"],"text":"A107131 \u2192 \u03b8+1"},"headline":"A107131 \u2192 \u03b8+1","chain":null}