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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":"18170fdb047e2231efb38c25f2cc7c3465466dce444a936d62679f3871663b66","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-a7e9ebb9860a9dbe","role":"sequence","name":"Stirling2 triangle with scaled diagonals (powers of 2).","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,      4;\n  2:   4,     24,        9;\n  3:   8,    112,      108,       16;\n  4:  16,    480,      900,      320,       25;\n  5:  32,   1984,     6480,     4160,      750,       36;\n  6:  64,   8064,    43344,    44800,    14000,     1512,      49;\n  7: 128,  32512,   278208,   435456,   210000,    38304,    2744,     64;\n  8: 256, 130560,  1742400,  3978240,  2780400,   762048,   90552,   4608,   81;\n  9: 512, 523264, 10748160, 34923520, 34020000, 13148352, 2304960, 192000, 7290, 100;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(8*x-9*x^2)*P(n-2,x) - (8*x-8*x^2-4*x^3)*d/dx P(n-2,x) - (4*x^2-8*x^3)*d^2/dx^2 P(n-2,x) + (2+4*x)*P(n-1,x) + (4*x-4*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 4*x;\nP[n_, x_] := P[n, x] = Expand[-(8*x - 9*x^2)*P[n - 2, x] - (8*x - 8*x^2 - 4*x^3)*D[P[n - 2, x], x] - (4*x^2 - 8*x^3)*D[P[n - 2, x], {x, 2}] + (2 + 4*x)*P[n - 1, x] + (4*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] = 2 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(8*x - 9*x^2)*v[i - 2] - (8*x - 8*x^2 - 4*x^3)*deriv(v[i - 2], x) - (4*x^2 - 8*x^3)*deriv(deriv(v[i - 2], x), x) + (2 + 4*x)*v[i - 1] + (4*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-a7e9ebb9860a9dbe","steps":["(\u03b8+1)\u00b2"],"edges":["e-d445ae4d7e7f6688"],"text":"A075497 \u2192 (\u03b8+1)\u00b2"},"headline":"A075497 \u2192 (\u03b8+1)\u00b2","chain":null}