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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":"4f4d1b7465db5d4714b3704dd016e11a4460349030385b43059729455f1c0aae","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-07af2e6e3c9f0c77","role":"sequence","name":"basis_map_A007318 of Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.","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:      3,       4;\n  2:     11,      30,       15;\n  3:     45,     186,      189,       56;\n  4:    197,    1092,     1680,     1008,      210;\n  5:    903,    6280,    12960,    11760,     4950,      792;\n  6:   4279,   35802,    92745,   112860,    71775,    23166,     3003;\n  7:  20793,  203350,   634095,   968660,   825825,   402402,   105105,    11440;\n  8: 103049, 1153416,  4206048,  7735728,  8278270,  5405400,  2130128,   466752,   43758;\n  9: 518859, 6540876, 27310608, 58753968, 75700170, 62034336, 32746896, 10820160, 2040714, 167960;\n","recurrence_sentence":"The row polynomials satisfy -(4*n+8-(n^3+3*n^2+2*n)*x^2)*P(n,x) = -(12*n+24+(4*n^2+20*n+24)*x-(3*n^3+9*n^2+6*n)*x^2-(2*n^3+8*n^2+10*n+4)*x^3)*P(n-1,x) - (8+12*x-(4*n^2+8*n)*x^2-(6*n^2+12*n+6)*x^3-(2*n^2+4*n+2)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(12*n + 24 + (4*n^2 + 20*n + 24)*x - (3*n^3 + 9*n^2 + 6*n)*x^2 - (2*n^3 + 8*n^2 + 10*n + 4)*x^3)*P[n - 1, x] - (8 + 12*x - (4*n^2 + 8*n)*x^2 - (6*n^2 + 12*n + 6)*x^3 - (2*n^2 + 4*n + 2)*x^4)*D[P[n - 1, x], x])/(-4*n - 8 + (n^3 + 3*n^2 + 2*n)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(12*n + 24 + (4*n^2 + 20*n + 24)*x - (3*n^3 + 9*n^2 + 6*n)*x^2 - (2*n^3 + 8*n^2 + 10*n + 4)*x^3)*v[i - 1] - (8 + 12*x - (4*n^2 + 8*n)*x^2 - (6*n^2 + 12*n + 6)*x^3 - (2*n^2 + 4*n + 2)*x^4)*deriv(v[i - 1], x))/(-4*n - 8 + (n^3 + 3*n^2 + 2*n)*x^2)); 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-66e2c3448245b0b7","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-190c283dd8b47413","e-15db7ce40c562b54"],"text":"A033282 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A033282 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}