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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":"7903bc925a74798f035a2a7ffb1743e29d14ba7ea8d8d59fde49d6b20a146801","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-b9fbfb57d9269e2f","role":"sequence","name":"Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= 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: 0,      4;\n  2: 0,      4,      9;\n  3: 0,      8,     27,      16;\n  4: 0,     24,     99,      96,      25;\n  5: 0,     96,    450,     560,     250,     36;\n  6: 0,    480,   2466,    3600,    2125,    540,     49;\n  7: 0,   2880,  15876,   25984,   18375,   6300,   1029,    64;\n  8: 0,  20160, 117612,  210112,  169225,  70560,  15778,  1792,   81;\n  9: 0, 161280, 986256, 1889984, 1682100, 808164, 222264, 34944, 2916, 100;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2-4*n+4+(4*n-8)*x-9*x^2)*P(n-2,x) + ((4*n-8)*x^2+4*x^3)*d/dx P(n-2,x) + (2*n-3+4*x)*P(n-1,x) - 4*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4*x;\nP[n_, x_] := P[n, x] = Expand[-(n^2 - 4*n + 4 + (4*n - 8)*x - 9*x^2)*P[n - 2, x] + ((4*n - 8)*x^2 + 4*x^3)*D[P[n - 2, x], x] + (2*n - 3 + 4*x)*P[n - 1, x] - 4*x^2*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 - 4*n + 4 + (4*n - 8)*x - 9*x^2)*v[i - 2] + ((4*n - 8)*x^2 + 4*x^3)*deriv(v[i - 2], x) + (2*n - 3 + 4*x)*v[i - 1] - 4*x^2*deriv(v[i - 1], 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-b9fbfb57d9269e2f","steps":["(\u03b8+1)\u00b2"],"edges":["e-c8880bf665496009"],"text":"A132393 \u2192 (\u03b8+1)\u00b2"},"headline":"A132393 \u2192 (\u03b8+1)\u00b2","chain":null}