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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":"ea198cc726dfafb79297c156519278f1a278b464de9f3d018de7ee33d8e3d3a8","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-a2b8ff03da0c14a3","role":"sequence","name":"Triangle read by rows: 3-Stirling numbers of the second kind.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:     1;\n   2:     3,      4;\n   3:     9,     28,       9;\n   4:    27,    148,     108,       16;\n   5:    81,    700,     873,      288,       25;\n   6:   243,   3124,    5940,     3280,      625,      36;\n   7:   729,  13468,   36729,    30240,     9500,    1188,     49;\n   8:  2187,  56788,  213948,   246736,   113750,   23184,   2058,    64;\n   9:  6561, 235900, 1197513,  1860768,  1181775,  349272,  50078,  3328,   81;\n  10: 19683, 969844, 6518340, 13293520, 11179875, 4495932, 926100, 98688, 5103, 100;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(6+12*x-9*x^2)*P(n-2,x) - (6*x-12*x^2-4*x^3)*d/dx P(n-2,x) - (x^2-4*x^3)*d^2/dx^2 P(n-2,x) + (5+4*x)*P(n-1,x) + (2*x-4*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 4*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 3 + 4*x;\nP[n_, x_] := P[n, x] = Expand[-(6 + 12*x - 9*x^2)*P[n - 2, x] - (6*x - 12*x^2 - 4*x^3)*D[P[n - 2, x], x] - (x^2 - 4*x^3)*D[P[n - 2, x], {x, 2}] + (5 + 4*x)*P[n - 1, x] + (2*x - 4*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + 4*x; for(n = 3, N, my(i = n); v[i] = -(6 + 12*x - 9*x^2)*v[i - 2] - (6*x - 12*x^2 - 4*x^3)*deriv(v[i - 2], x) - (x^2 - 4*x^3)*deriv(deriv(v[i - 2], x), x) + (5 + 4*x)*v[i - 1] + (2*x - 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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-a2b8ff03da0c14a3","steps":["(\u03b8+1)\u00b2"],"edges":["e-0cb5aa2caeb40953"],"text":"A143495 \u2192 (\u03b8+1)\u00b2"},"headline":"A143495 \u2192 (\u03b8+1)\u00b2","chain":null}