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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":"afbcd7069836fc7db763a04288d322796877e60a4c718f98efb04bf75eae99e3","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-67ad51294793de36","role":"sequence","name":"basis_map_A007318 of Triangle T(n, k) read by rows: T(n, k) = Sum_{j=0..n} binomial(j, n-k) * |Stirling1(n, n-j)|.","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:        1,         2;\n  2:        3,        10,          6;\n  3:       15,        68,         75,         24;\n  4:      105,       596,        933,        568,        120;\n  5:      945,      6414,      12867,      11332,       4630,        720;\n  6:    10395,     81894,     199263,     227588,     135920,      41256,       5040;\n  7:   135135,   1210152,    3450306,    4818432,    3758355,    1678056,     402444,     40320;\n  8:  2027025,  20314440,   66276414,  109079744,  104559645,   61251048,   21698516,   4284288,   362880;\n  9: 34459425, 381831930, 1400943978, 2649672616, 3004661085, 2170511982, 1012010776, 296017632, 49547376, 3628800;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = -(4*n^3-12*n^2+9*n+(4*n^3-10*n^2+6*n)*x+(n^3-2*n^2+n)*x^2)*P(n-2,x) + (4*n^2-6*n+1+(2*n^2-2*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 12*n^2 + 9*n + (4*n^3 - 10*n^2 + 6*n)*x + (n^3 - 2*n^2 + n)*x^2)*P[n - 2, x] + (4*n^2 - 6*n + 1 + (2*n^2 - 2*n)*x)*P[n - 1, x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^3 - 12*n^2 + 9*n + (4*n^3 - 10*n^2 + 6*n)*x + (n^3 - 2*n^2 + n)*x^2)*v[i - 2] + (4*n^2 - 6*n + 1 + (2*n^2 - 2*n)*x)*v[i - 1])/(n - 1)); 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-849de7332df28657","steps":["basis A007318","\u03b8+1"],"edges":["e-34da37650456524d","e-065fd5489dc9ea2f"],"text":"A088996 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"A088996 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}