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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":"9ea0d18977a1ac2dc70f79eaa4a85c712195ab52aaba8760af56a855c9c82193","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-df97cd413ec843e8","role":"sequence","name":"Triangle read by rows, the Bell transform of the triple factorial numbers A007559(n+1) without column 0.","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:           4,           4;\n   3:          28,          48,           9;\n   4:         280,         640,         216,          16;\n   5:        3640,       10080,        4680,         640,         25;\n   6:       58240,      185920,      106920,       20480,       1500,        36;\n   7:     1106560,     3951360,     2663640,      640640,      66500,      3024,       49;\n   8:    24344320,    95307520,    72817920,    20751360,    2730000,    177408,     5488,     64;\n   9:   608608000,  2575372800,  2180525760,   710492160,  111594000,   9253440,   411600,   9216,    81;\n  10: 17041024000, 77120243200, 71210361600, 25901747200, 4681152000, 466058880, 26589360, 860160, 14580, 100;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9*n^2-33*n+30+(12*n-20)*x-9*x^2)*P(n-2,x) - ((6*n-10)*x-(12*n-20)*x^2-4*x^3)*d/dx P(n-2,x) - (x^2-4*x^3)*d^2/dx^2 P(n-2,x) + (6*n-8+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) = 4 + 4*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 4 + 4*x;\nP[n_, x_] := P[n, x] = Expand[-(9*n^2 - 33*n + 30 + (12*n - 20)*x - 9*x^2)*P[n - 2, x] - ((6*n - 10)*x - (12*n - 20)*x^2 - 4*x^3)*D[P[n - 2, x], x] - (x^2 - 4*x^3)*D[P[n - 2, x], {x, 2}] + (6*n - 8 + 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] = 4 + 4*x; for(n = 3, N, my(i = n); v[i] = -(9*n^2 - 33*n + 30 + (12*n - 20)*x - 9*x^2)*v[i - 2] - ((6*n - 10)*x - (12*n - 20)*x^2 - 4*x^3)*deriv(v[i - 2], x) - (x^2 - 4*x^3)*deriv(deriv(v[i - 2], x), x) + (6*n - 8 + 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-df97cd413ec843e8","steps":["(\u03b8+1)\u00b2"],"edges":["e-d3c79fa9007149b0"],"text":"A035469 \u2192 (\u03b8+1)\u00b2"},"headline":"A035469 \u2192 (\u03b8+1)\u00b2","chain":null}