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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":"e3cd972b4012dda539c290d12b1fa6a1c262d031cdf1660fbfa55b397c14e76a","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-2887a9a5b94917e3","role":"sequence","name":"basis_map_A007318 of Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.","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,       6;\n   3:     16,      40,      15;\n   4:     64,     224,     168,       28;\n   5:    256,    1152,    1296,      480,      45;\n   6:   1024,    5632,    8448,     4928,    1100,      66;\n   7:   4096,   26624,   49920,    39936,   14560,    2184,      91;\n   8:  16384,  122880,  276480,   281600,  144000,   36288,    3920,    120;\n   9:  65536,  557056, 1462272,  1810432, 1196800,  430848,   79968,   6528,   153;\n  10: 262144, 2490368, 7471104, 10895360, 8852480, 4173312, 1123584, 160512, 10260, 190;\n","recurrence_sentence":"The row polynomials satisfy (8*n^2-20*n+12+(4*n^3-12*n^2+11*n-3)*x)*P(n,x) = (32*n^2-80*n+48+(16*n^3-76*n+60)*x+(12*n^3-20*n^2-5*n+12)*x^2)*P(n-1,x) - ((16*n-16)*x+(8*n^2+16*n-22)*x^2+(8*n^2-6)*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((32*n^2 - 80*n + 48 + (16*n^3 - 76*n + 60)*x + (12*n^3 - 20*n^2 - 5*n + 12)*x^2)*P[n - 1, x] - ((16*n - 16)*x + (8*n^2 + 16*n - 22)*x^2 + (8*n^2 - 6)*x^3)*D[P[n - 1, x], x])/(8*n^2 - 20*n + 12 + (4*n^3 - 12*n^2 + 11*n - 3)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((32*n^2 - 80*n + 48 + (16*n^3 - 76*n + 60)*x + (12*n^3 - 20*n^2 - 5*n + 12)*x^2)*v[i - 1] - ((16*n - 16)*x + (8*n^2 + 16*n - 22)*x^2 + (8*n^2 - 6)*x^3)*deriv(v[i - 1], x))/(8*n^2 - 20*n + 12 + (4*n^3 - 12*n^2 + 11*n - 3)*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-ba8de4e02acfb6ca","steps":["basis A007318","\u03b8+1"],"edges":["e-56956d6724ecd488","e-4c6ed80a48a5cf51"],"text":"A091042 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"A091042 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}